Part 8
II
CONTENTS
42.1 Symbolic computation 919
42.2 Projects 920
Symbolic computation
There have been a number of significant advances in symbolic computation and computer algebra manipulation in recent years. These are systems which bring together symbolic, numerical, and graphical operations in one software package. The mathematical methods introduced in this book are particularly appropriate contexts in which to have a first look at such systems.
The software Mathematicaf has been used extensively in the production of the drawings of curves and surfaces, and in the checking of examples and problems, in this text. At an elementary level, Mathematica is particularly helpful, for example, with operations such as differentiation (including partial derivatives), the construction of Taylor series, elementary algebraic operations involving matrices and linear equations, elementary integration (including repeated integrals), and difference equations; but most topics in this book can be approached to some extent using Mathematica. It is also useful in curve sketching in that a quick view of the general feature of a curve can be obtained, which can then be revised and edited to produce detailed graphs as required.
It is not the purpose of this book to provide an introduction to Mathematica. There are a number of texts which do, including the handbook that comes with the system. There arc other software packages including MAPLF/1 which can also be used in mathematics. Apart from this chapter, Mathematical Techniques is software-free.
Useful information about Mathematica and its applications can be found in the following texts by Abell and Braselton (1992), Blackman (1992), Skeel and Keeper (1993) and Wolfram (1996).
f Mathematica is a registered trade mark of Wolfram Research In + M APLE is a registered trademark of Waterloo Maple Software.
2.2
Projects
The following projects are listed by chapter. They are selected samples of problems and do not cover every topic in the book. The intention is that they can be approached using mainly built-in Mathematica commands: very few problems require programming in Mathematica. It is generally inadvisable to attempt these problems by hand, since many could involve a great deal of manipulation, although some projects are prompted by examples and problems in the relevant chapters.
It is worth emphasizing that computer algebra systems usually generate outputs or answers without explanation of how the results are arrived at, unless the programming within them is investigated. Outputs can go wrong for many mathematical reasons. For example, a curve can oscillate too frequently for the built-in point spacing to detect, which can result in a false graph. This can be corrected by increasing the number of plot points, but the potential difficulty has to be recognized at the formulation stage. Symbolic computation is not a substitute for understanding mathematical techniques.
Mathematica notebooks for each project are available on the web at:
w w w. oxfo rdtcxtbooks. co. uk/ore/ j ordan_smith4e Any comments should be sent to the authors at:
School of Computing and Mathematics, Keele University, Keele, Staffordshire ST5 5BG, UK. (Email: p.smith@maths.keele.ac.uk)
Chapter 1
1. Draw the graphs of y = xi, y = (x - 1)\
y - 1 = x\ y - 1 = (x - 1 )•' for -1.5 ': x =s 2.5. How do they differ?
2. (a) Plot the points («, nl + 1) for n= 1, 2, 3, 4, 5.
(b) Plot the points in (a) but with sueeessive points joined by straight lines.
(c) Plot y = x1 between x = 0 and x = 5.
(d) Show the curves from (b) and (c) on the same graph.
3. Plot curves defined by the following relations between x and y.
(a) x2 + 3y2 = 4; -2«Sx«2;
(b) x1 + 2y1-xy + 2y = 4; -3 =S x =5 3;
(c) x4 + 2yl - xy - 2xy = 4;   -2 « x 3.
4. Define the function f(x) = x(l — x1). Plot the graphs
(a) y = f{x);
(b) y = f(l -x);
(c) y = f(-xh
(d) y = f{\x\); all for-2 *Sx =S 2.
5. Define the Hcavisidc function H(t) and the signum function sgn t. Plot graphs of the following functions on -4 =£ t =£ 4:
(a) H(t);
(b) sgn/;
(c) HW+Hi-t);
(d) sgn(stni).
6. Plot the graphs of the curves defined by the following polar equations:
(a) r= 7(1 -cos 8) forO s= 0s£ n (cardioid).
(b) r = (4 sin20- 1) cos 6 for 0 =S 0 < 2% (folium).
7. Express
_1_
(x--\)(x-2)(x-3)(x-4)(x-5)
in partial fractions.
Chapter 2
1. Define the function
x sin x — 1 + cos x
sin 2x + 2 - 2 er
Find limv^o f(x). Plot the function fot —0.5 x =S -0.001 and fot 0.001 « x =S 0.5, and check gtaphically that this agtecs with the limit.
2. Find the derivative of
fix) = Ix1 + 8x% + 9x4 + I0x5 + 1 Ix6 + 12x7 and its values /"'(0.2) and f (0.4).
3. Find the derivative of
f{x) = x4 + 2xi - 3.Y2 - Ix + 4.
Find the approximate values of x whete f'(x) =0, using a numerical solution routine. Plot graphs of y = f(x) and y = fix) on the same axes and compare the zeros of fix) with the zero slopes on y = f(x).
4. Find the equation of the tangent to the curve
y = x sin 2x
at x = 0.7. Plot the graphs of the curve and irs tangent.
5. Find the first three derivatives of
f(x) = x sin2* + x1 sin (x2),
and confirm that the first nonzero higher derivative at x = 0 is fm(0) = 6.
6. Plot the graphs of y = f(x), y = /'(x), and y = f"{x) for
f(x)=x1{x1-'i)
in rhc interval -2 =S x =£ 2.5. (This should confirm the results from Problem 2.1.9.)
Chapter 3
1. Display rules for the derivatives of the following general forms:
(a) f(x)g(x);
(b) f(x)/g(x); (C) f(g(x));
(d) f(x)g(x)h(x);
(e) f(x)g(x)/h(x);
(f) f(h(x))/b(x).
2. Find rhe first derivatives of
f(x) — smx'x sin x.
The function is periodic. What is its minimum period? Plot its graph and the gtaph off'(x) over one cycle. Estimate where f(x) is stationary and then find each of the roots of f'(x) = 0 to 5 decimal places using a root-finding routine.
3. If
x1 + ly1 — xy — 2y.r2 = 4, find dy I Ax as a function of x and y.
Chapter 4
1. Display rules for the first and second derivatives with respect to x of the following general forms:
(a) f(x2);
(b) f(sinx);
(c) fismix1)).
2. Find the first and second derivatives of
f(x) = 0.1 x" - 0.5x4 + 0.2.x-' + x- - 0.7.v + 2.2.
Estimate the roots of f'{x) = 0 from a graph of y = f(x). Then find the roots to 5 decimal places by a root-finding routine. Calculate f"{x) at each stationary point, and confirm the second-derivative test for stationary points. Points of inflection are given by f"(x) = 0. Find their locations on the original graph of y = f(x).
3. Plot the graph of
x2-1
2x+\
and its asymptotes y = \x - \ and x = —{ (see Fig. 4.13).
4. Plot the graph of y = f(x) = x5 - Ix3 + x2 - ix + 1 in the interval — 1 =S x =£ 3, and estimate the roots of f{x) = 0 in this interval. Set up a Newton routine
x„+l = x„~j-^-,
for calculating the roots of f(x) = 0, and find, starting at x = 0.5 and 1.6, the roots to 10 significant figures. What is the smallest number of iterations required in each case to calculate the roots to 10 significant figures?
5. Plot the graph of y = x + sin Sx in the interval 0    x =S 25 using
(a) the default plotting routine,
(b) plotting with 20 plot points,
(c) plotting with 50 plot points.
Explain why the graphs are different for this type of function.
Chapter 5
1. Obtain formulae for the Taylor polynomials for the following functions ccntted at x = a as far
as (x - a)3:
(a) f{x); (b) [f(x)Y;
(C)  f(x)g(x);       (d) e«*i.
State the coefficient of (x — a)1 in each case.
o
c
m
O
H
«
C3 2
□ a.
O
o
Ü
o
CO
S
>
CO
(D Z
CO
CO h-
o
UJ ~5
o
DC Q.
CO
< Ü _i
a a.
<
2. Find Taylor expansions about x = 0 up to and including    for each of the following functions: (a) e1; (b) (x+l)cosx;
(c) ln(l + sin.T);       (d) exp(sin(ex-1)).
3. Find the Taylor polynomials for (n'm1x)/x2 up to and including xN for N = 2,4, 6. Plot the graphs of the function and its Taylor polynomials
for 0.001 SiS2, and compare them. At approximately what values of x do the Taylor polynomials visibly part company from the exact function?
4. Find the Taylor polynomials for lnx about x = 1 for N = 6. Construct an error function which is the difference of In x and its Taylor polynomial. Show that, at 2.159 approximately, this error starts to exceed 0.2 as x increases. Plot this error function against x for 1 =S x =£ 2.2.
Chapter 6
1. Solve, for the complex number a, the equation z~0 where
.= (2 + .3i)< | (a-2a)
2,
5.
(l-5i)3 (l+5i)4
If z — x + iy, find the real and imaginary parts of z tr cos z.
Find the 13 roots of z13 = 1 + i, and plot the roots on the Argand diagram.
Let 2j = 1 - 2i, z2 = 3 + i. Plot the following points on the Argand diagram:
Z\ "t" Z?,   Z-[ "t" Z2,   Z[    -C2,   Z\ ~t~ Z2,   Z]Z2, Zj/z2.
Find I z I and Arg z, where _ (l + 2i)4   2(3-4i)3
(l + 3i)
l + 4i
Chapter 7
1. Let
C =
I	2	3	4
-2	3	-4	1
3	4	1	2
4	-1	2	3
1	0	-1	o"
1	-2	1	2
-3	1	-3	1
2	1	2	1
"3	1	2	r
		1	2
1	-2	-3	2
2	1	0	-1
Find and compare
(a) AB and BA;   (b) A(BC) and (AB)C;
(c) (A + B)T and AT + BT;   (d) (AB)T and BTAT.
2. Find the inverse of
1 *, 1 x2 1 x,
(see Problem 7.18). Find the equation of the parabola of the form y = a + bx + cx1 through the points (-1, -2), {\, -1), and (f, 2).
3. Let
Find A2, A4, A8, A16. How do you expect A" to behave as n —> °°?
Chapter 8
1. Let
1	-I	2 3"		
3	1	0 -3		
2	-1	3 -1		
2	-1	2 4		
2	4	-3	l"	
0	-1	4	3	
-2	-2	3	1	
-2	5	6 -	-5	
Find det A, det B, det A ', and det AB. Confirm that
det A"1 = 1/dct A, det A det B = det AB. 2. Factorize the following determinants:
	1	1	1	
a)	a	b	c	
	a1	b1	c1	
	1	1	1	1
c)	a	b	t:	d
	a1	b1	2 C	d1
	aA	b4	c4	d4
(b)
1111
abed a1 b2 c2 d2 a3   b3 c3
3. Find the values of a for which -1
Chapter 9
1. Plot the curve which has the position vector
r= (2 cos t)i+ (2 sin t)j + 03tk
from t = 0 to t = 20. What is the curve called? The position vector represents a particle moving along the curve. Find the velocity vector r and the acceleration vector r of the particle. Show that r-r— 0.
2. Plot the trefoil knol given parametrically by
r= {1 + a cos 32) (cos 2t i+ sin 2tj)+a sin 3f fe with a ~ 0.25 and 0 < t « 2lt.
Chapter 10
1. Show that
_2-3/2 2-1/2 2-;i/23l/: -2-312 _2-i« 2-,/23I/: 2-131/2     0 2"1
defines a rotation of axes. If each row defines the direction of the X, Y, Z axes in the x, y, z frame, find the equation of the plane x + 2y — 2z — 1 in the new axes.
Chapter 11
1. The area of a triangle whose vertices are the points with position vectors a, b, and c is given by the formula
\\b Xc + cXa + axb\.
Devise a program based on this formula to determine the area for general vertices. What is the area if«=(l,0, 1), fe = (2, -1, 1), and e= (1,1,2)? Plot a diagram showing the triangle.
2. A tetrahedron has vertices with position vectors
«=(1,-1,2),   * = (-1.2,3),
c= (2,-1,3),   d= (1,3,-2). Find its surface area. Draw a three-dimensional plot showing the tetrahedron viewed from the point with position vector (2.1, -2.4, 1.5).
Chapter 12
1. Use a row-reduction routine to solve the linear equations
x + 2y — 3z = q,
2x + py+  z = -l,
x — 2y— z = 4,
where p and q are two parameters. Determine for what values of p and q the equations have (a) a unique solution, (b) no solution, (c) an infinite set of solutions.
2. Use a row-reduction method to solve the linear equations
x + 2y + pz = 5,
3x + 2y + z = q,
2x - y + 4z = 7,
where p and   are two parameters. Confirm that
63 — 5o    ,       I,. ll + 7p '
and discuss the nature of solutions for all values of p and q.
3. Using a row-reduction instruction, show that
x1 + 3%      = 5,
—xt+ x2—    Xj + x4 = —l, X, +2*2+11*3        = 4,
—x, + 2%, + 3*3 + x,, = 3 is an inconsistent set of equations.
Chapter 13
1. Find the eigenvalues and eigenvectors of
to
TJ T! O c_
m o
H
(73
-6 1 2
1 0 -3
2 1 -6 -2 2 0
How many linearly independent eigenvectors does A have?
Find the eigenvalues of the following matrices:
(a) A-1;   (b) A2;   (c) A + kl. 2. Find the eigenvalues and eigenvectors of
1 2 l]
2 11 1   1 2
Construct a matrix C of eigenvecrors and confirm that
/\ = CDC~\
where D is a diagonal matrix of eigenvalues. Obtain the general formula for
A" = CD'CK 3. Find the inverse and Transpose of
A =
and verify that A is an orthogonal matrix. Find the eigenvalues of A. What expected property do thev have?
4. Find the eigenvalues of
O Z
h-
D 0.
5 O
o Ü
_1
o
CO
>
CO O
CO 3
oo
h-
O
UJ
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O
rr
0.
to
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a,
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5 -3
-3
5 13
7
3 -15
-6 -6 0 12
Find the expression det(A - /ll4), and demonstrate the Cayley—Hamilton theorem of Problem 13.21.
Chapter 14
1. Plot the graphs of the derivative dy/dx = sin 2x and the equation of the curve through (71, -1) of which this is the derivative (see Example 14.7).
2. Plot the graph of
-f- = x c  + sin x — x- cos 2x, dx
for 0 =S x =S 10. Show that an antiderivative which is zero when x = 0 is
y = 2 + f [-4(1 + x) sT* - 4 cos x - 2x cos 2x
+ sin 2x — 2x2 sin 2x].
Plot the graph of the signed area between x = 0 and x = 10.
Chapter 15
1. Set up a program to compute the area under the curve y = f(x) between x = a and x = b using the approximation
where h = (b — a)/N and x„ = a + nh. Apply the method to the following functions, limits, and subdivision numbers:
(a) f{x) = x\ 1 «xs=3,N = 200;
(b) f{x) = x e-', 0 «S x =S S, N = 20;
(c) fix) = x3 sin x, 0Sx«n,N = 30;
(d) fix) = cos(c-'), 0 «S x =S 1, N = 25.
In cases (a), (b), and (c), compare rhe numerical result with the areas obtained by integration. In these cases, how many subdivisions are required to obtain a numerical result correct to 3 decimal places? In (a), show that over 10 000 steps are required. Why is this?
2. Use a symbolic integration program to obtain the following indefinite integrals:
(a) (lnx)!dx;
(c)     x2 c* sin x dx;
(b)
(d)
sin5x cos5x dx;
"v(l -x2) dx;
(e)
(f)
dx
x(x+ 1)(x + 2)(x + 3)' dx
(1-X-'
Check each answer by recovering the integrands by differentiation.
3. Evaluate the following definite integrals:
x dx
(a)   I x(lnx)Jdx; (b) x' dx
(d)
0 V(5 + 4x-4x2)'
"1 100
S^rdx.
4. Find
1(a) = (lnxfdx. Ji
Find the limit
lim^M-. h^o (-In by
How does Ha) behave as a —> °°? Does
(In x)3 dx exist?
5. A cylindrical hole of circular cross-section and
radius b is drilled through a sphere of radius a> b, the axis of the hole passing through the centre of the sphere. Find the volume of the remaining object. Display a diagram of the object for some values of a and b.
Chapter 16
1. Plot the graph of the polar equation r = sin 50 for 0 =S 9 =S 271. Find the area enclosed by the five 'petals' of the curve.
Show that the area of the 2n + l petals of r= sin(2« + 1)0(nS? 1) is independent of n.
2. Devise a program to generate the trapezium rule:
f(x) dx-^f-[\M + (f(x,) + f(x2) + ■
Apply the program to the integral
f 2
e 2x sin2x dx,
and compare the result with the exact value of the integral. Investigate how many steps are
required to obtain a result accurate to 3 decimal places.
Apply the program also to Problem 16.20.
3. A thin plane metal plate consists of an isosceles triangle of height /; and base length 2a with a semicircle of radius a attached symmetrically by its diameter to the base of the triangle. Find the location of its ccntroid on its axis of symmetry.
4. Set up a program to generate Simpson's rule
f(x) dx ■
3N
-\f(a)+f(b)+4^f(xlkl)+2^f(x2k)
where N is an even number. Apply the method to f(x) = e , with b = 1, a = 0. Compare results with the trapezium rule above.
Chapter 17
1. Illustrate the substitution method in integration by writing a program to integrate
dx,
V(5 + 4x-x2)
using the substitutions x = n + 2, u = 3 sin f. Integrate directly and through the substitutions.
2. Integrate the following, and compare your answers with computer-integrated ones:
(a)
x dx 4^+1
(b)
(c)     cos4xdx; (d)
tan x dx;
x dx V(x-1);
-dx.
(e)
3. Computer-integrate the infinite integrals
twe-'dt,   In= tne'dt,
and confirm that 1U/IW = I i.
4. Computer-integrate the following infinite integrals:
| ^^-dx;
(c)
x' e dx
5. Evaluate the integral (In x
f(a)'-
dx
for a > 1. Find f{W), f(20), and f[<*>). The results indicate that f(a) tends to a limit very slowly as a —> <*>. Find where
g[x)
(In x)'
has a maximum value, and plot the graph y=g(x) for 1 =£ x =S 100.
Chapter 18
1. Solve the differential equation x + x = 0, for the initial conditions (a) x(0) = 0, (b) x(0) = 1, (c) x(0) = 2, and plot the solutions on the same axes for 0 =S t =s 2.
2. Solve the differential equations
(a) 2x + 3x + x = 0,      (b) x + 2x + 2x = 0, (c) x + 2x + x = 0,
each for the six sets of initial conditions:
(i) x(0)=0,x(0) = l;
(ii) x(0)=0,x(0) = 2;
(iii) x(0)=0,x(0)=3;
(iv) x(0) = 0,x(0) = l; (v) x(0) = 0,x(0)=2;
(vi)  x(0) = 0,x(0)=3.
Plot all solutions on the same axes for each differential equation, for 0 =£ t < 5.
Chapter 19
1. Solve the differential equation 2x + 3x + x = cos t subject to x(0) = 0, x(0) = 1. Plot the solution for 0 =£ t 50.
2. Solve the differential equation x + x— cos t subject to x(0) = 0, x(0) = 0. Plot the solution for 0 =S t =S 20.
Chapter 20
1. Solve the differential equation x + x = 0 subject to the initial conditions x(0) = I, x(0) = 0. Also solve x + sin x = 0, by a built-in numerical solution method for 0 =S t ^ 10 subject to the same initial conditions. Plot both solutions for 0 s t $ 10. Comparison of the plotted solutions will indicate by how much the period decreases when the linear approximation is used. Rerun the programs for different amplitudes x(0).
Chapter 21
1. Draw the phasor diagram of the sum of the three phasors of
a z
L. D
a.
O O
o _l o
5
>
CO
a
2
CO Z)
CO h-
o
lu
o
CC 0-
< o _l
CL
<
u(t) = 2 cos Wt,   v(t) = cos(10i - jrt), w{t) =3 cos(10f+|s) (see Example 21.6).
Chapter 22
1. Draw the lineal-element diagram of dyldx = xy, produced by a standard package in the square jOSx« 1,0=S>'=S 1} (seeSection22.1). Compare this with the exact solutions (see Section 22.1) drawn through the points (0, 0.2), (0,0.4), and (0, 0.6).
2. Repeat the above process for the differential equation dy/dx = x - y of Example 22.1.
3. Design a program for Euler's method (Section 22.2) for the initial-value problem
dy: dx
Chapter 24
2.
-■xy
y(0) = i
(see Example 22.4) with step length h - 0.2 and five steps. Run the program for the cases h = 0.1 and h = 0.01 and compare the results.
4. Plot numerical solutions for
dZ; dx
3y
3x -
y
(Example 22.14 and Fig. 22.11) using built-in routines. As with many equations of this type it is often easier to solve the equivalent simultaneous equations
dx   „ dy
dT3*-* dT
= 3y - x,
numerically for various initial values of x(0) and y(0).
Chapter 23
1. By splitting the differential equation x + Ix1 = 0 into the system
x = y,      y = —Ix3,
and plotting four phase paths respectively through the four points
(x(0), y(0)) = (0.3, 0), (0.6, 0), (0.9, 0), (1.2, 0)
over the interval —1.5 =£ x =£ 1.5, show that the solutions appear to be periodic.
2. Plot phase paths for the van der Pol equation
x + mx2-\)x + x = 0
showing the limit cycle. Also show the corresponding (t, x) graph of the periodic solution (the periodic solution has an initial value close to x(0) = 2, x(0) = 0).
1.
Computer algebra systems arc quite efficient at finding Laplace transforms of complicated expressions involving standard functions. Test the system with the following transforms:
(a) L{t*c>};
(b) Lit1 e-' cos l};
(c) L
df
if 0 « f if t > c:
(d) L{f(t)} where f(t) =
(e) L{e'/t[};
(f) L{cos\\at).
2. Solve
x + 2x = e-', x{0)=3,
using a Laplace-transform package, and compare the answer with that of Example 24.12. Plot the input e ' and the output against t for
3. Using a Laplace-transform package, solve the system
x + 2x + x = a cos cot,      x(0) = 0, x(0) =0.
Plot the input and output functions for a = 1, (0= 1, and 0 «S t 30. Estimate the eventual amplitude of the periodic output.
4. Find the functions whose Laplace transforms are:
(a)
s{s+l){s + 2){s + 3) Plot the functions in each case
(b)
(s2 + 4)(5+l)'
5. Consider the function f(L) = In t. Show the Laplace-transform package produces the transform
(y + lns),
where J is Euler's constant given by 7= I'm ( Xi-ln
Derive a program to calculate Euler's constant. It should give 7=0.577 215... .
Chapter 25
1. Find the Laplace transform of the solution of
x + co2x-ao{t-l),      x(0) =x(0) = 0,
which has impulse input applied at time t=l. Invert the transform and plot the output for (0=4, a = \ (see Example 25.3).
C
1,
2. Following the previous project, solve the more complicated problem with two impulses:
2x + 3x + 2x = a B(t- n) cos t + b 8{t-2k),
x(0)=x(0) = 0.
Plot the output for a = b = 1.
3. Let f(t) = tl, g(t) = cos t. Find the convolution
t
f(t-u)g(u) du.
Then verify that L{f(t)}L{g(t)}=L
f(t - u)g(u) du
4. A transfer function with a parameter a is given by (Section 25.10)
, 4z3 - 8z2 - 2z + 4
(j(z) = —-:-;-;---.
6z4 — 6z3 — 2a~z~ + 3z2 + 2a1z — 2a1
Find the locations of the poles of Q(z). For what values of a do all poles lie within the unit circle (indicating transient stability) ? Plot the poles on an Argand diagram for a = 2.
Chapter 26
1. Consider the period-2 sawtooth function defined over its fundamental interval -1< t «S 1 by f(t) = t. Find its general Fourier coefficient and output its first four terms. Plot and compare the graphs of this truncated series and the sawtooth for -3 < t« 3.
2. Repeat the previous problem but with the function
fit)
1 (0<t<l), -l (-l=st<0).
Plot the graphs of f(t) and the first 12 terms of its Fourier series. The graph should show the Gibbs' phenomenon, in which the Fourier series approximation overshoots the function at discontinuities. You can try it with (say) 20 terms or more, but you should include more interpolating points in these cases.
3. Find the Fourier coefficients of the 271-periodic function defined by
f{x) =x6~ 5jt2x4 + 7rcV
on the interval —71 =S x < 7t. What is the sum of the series
v(-!)"+\
Chapter 27
1. Find the Fourier transforms of the following functions:
(a) the top-hat function 11(f);
(b) the one-sided exponential e"'H(t); (C) e-l'l;
(d) eH*"1';
(e) 1/(1 + 1*).
Plot the graph of the transform in (e).
2. Find the functions whose Fourier transforms arc
(a) e-'2;
(b) 1/(4 + /*);
(c) 2;
(d) 2cos(/-a). Chapter 28
1. Plot the saddle surface z = x1 — y2 in the cylinder x2 + y1 =£ 1, using a three-dimensional paramerric plot routine with parameters r
and u where
(x, y, z) = (r cos u,   rsinu,   r1 cos 2m).
Also draw a contour plot of the surface in the (x, y) plane on the square — 1 x =S 1, -1 =£ y =S 1.
2. Plot the surface z = xy(x2 — >'2) in the cylinder x2 + y1 *S: 1 using the same routine as in Project 28.1 above, but with the parametric equations
(x, y, z) = [r cos u, r sin u, \r3 sin 4a).
How would you describe this saddle? Draw-its contour plot in the square — 1 S x 1, -1 « y « 1.
3. For the function
f(x, y) =try sin(xy) + x ln(x2 + y3), verify that
y/ = sj _
dy dx dxdy
4. Plot the surface given by z = cos xy over -71 =S x =S 71,--jll =S y s£ |jt. Find the partial derivatives at (\n, 1) and construct the equation of the tangent plane there. Finally plot the surface and its tangent plane.
5. Find the stationary points of
f(x, y) = 0.3x3 + 0.2y2 - xy - xy + 2y numerically by solving
^ = ^ = 0. dx dy
Plot the contours on the (x, y) plane for -3 « x =S 3,-9 =S y =S 3.
Find the values of the second derivatives at each stationary point and check the second derivative tests (28.9) at each point.
6. Find the least-squares straight line fit to the points
(0,1.1),   (1,2),   (2,2.9), (3,3.9), (4,4.5), (5,5.1),
in the [x, y) plane. Plot the data and the least-squares straight line fit. If you are using a built-in routine, check your results against that given by (28.10).
Chapter 29
1. Find the family of curves orthogonal to that of dy
— = ye*. dx
Plot both families of curves for |x| ^ 2, \y =S 2.
Chapter 30
1. Find where the function
f(x, y) = x3 — 2xy — x + iy2
is stationary subject to the condition x1 + 2y2 = 1. Devise a program which uses the Lagrange-multiplier method (30.4): here is a suggested line of approach. First plot the contours of z = f(x, y) and the curve x1 + 2y2 = 1. Locate the approximate coordinates of any point of tangency. Then use a built-in root-finding scheme to locate the stationary values. There should be four.
Chapter 31
1. Find the equation of the tangent plane to the surface
x'y + zx + xy2z = -3
at (1,2,-1).
2. Show graphically the intersection of the cylinder x1 + y1 = 1 and the plane x + y + z = 1 (Example 31.9).
3. Find the envelope of the family of curves
y(a2 — 1 + ax) = x
with parameter a. Plot the envelope and a sample of touching curves in — 3 =S x =§ 3.
Chapter 32
1. By repeated integration, evaluate the integral ■1 ri
(x + y t~xy + xy) dx dy,
J-i Jo
using a symbolic routine. Plot the surface
z = x + y e "y + xy
over 0    x    1, -1 =£ y =£ 1. Interpret the integral as the volume under the surface. Does the integral contain 'negative' volumes under the surface? Plot the positive part of the surface over the same rectangle.
2. Evaluate the repeated integral x2)1 dx dy.
Plot the region of integration in the (x, y) plane, and then check that the integral has the same value with the order of the integration reversed.
Chapter 33
1. Let
f{x, y, z) = xyi + yzj + (z - y)xk.
Find /"as a function of t on the line x = t, y = t, z-t. Evaluate the line integral
f-dr
on this line between (0, 0, 0) and (1, 1, 1).
Repeat the process with the curve x = f2, y = t\ z = t4, and the same end-points. Plot both paths of integration.
Chapter 34
1. Plot the surfaces defined paramctrically by the following position vectors:
(a) r = (3 + cos v) cos u i + (3 + cos v) sin u f + sin v k (see Section 34.3);
(b) r=(l + a sin(bu)) cos v i+ (1 + a sm(bu)) sin v j + uk, where a = 0.3 and b = 3.5 (see Section 34.3).
2. Given that
f(x, y, z) = cxy'i + z cos(xy); + (x2 + y2)k, find
(a) dxvfi
(b) curl/";
(c) div curl f\
(d) curl curl f at the point (1, 0, -1).
3. Using symbolic computation test the validity of the following identities:
(a) (F■ grad)F = ^grad(F-F)-FxcurlF;
(b) div(fXG)=b-curlF-F-curlG;
(c) curl(F X G) = (G-grad)F - (F-grad)G - G div F + F div G;
(d) div(U grad V-V grad U) = UV2 V - VV2[J;
(e) curl curl F = grad div F — V2F.
Chapter 35
1. A and B are the sets of integers defined by
A = {2W + 5(-l)"|neM+, 1 =S « =s 100},
B = {n2-n + l\neK\ 1«»« 10}.
Produce lists of the elements in A U B and A Pi B. How many elements do each of these sets have?
2. Let A, B, and C be the following sets:
A = {«(n-l)|neN*,2« « =S 100},
B= {|rc2-100«J |«eN+, I ««=£160},
C={4«|neN+, l=Srcs=220<)}.
Verify the first distributive law
An (flu C) = (AnB)u(An C).
How many elements are there in the set An(BkjA)>
Chapter 36
1. Design programs to generate the truth tables for the or gate, the and gate, the not gate, the nand gate, and the nor gate.
2. Design a program to simulate the truth table in Example 36.3 which has the output
f = a * b © b © c
for inputs a, b, and c.
Chapter 37
1. In the cutset method applied to the circuit in Fig. 37.23, the currents iv i2,i3, <4, «5 and the voltages u„ vb, vc, vd satisfy the nine equations
i1 — i3 + (2 = 0,       ix - i3 + i2 = 0,
—iY + i5 — (3 + i2 = 0,
-!y + !4 + i2 = 0,
H={vc-vh)IKz, i. = (vh-vd)/R., U=(vc-Vj)IKA, h = v„/R5,
where ix = 2 A, iy = 2 A, and R, = \Q., R, = 3 il, R, = 1 il, R4 = 2 £2, R5 = 2 £2. Solve this set of linear equarions for the currents and voltages.
2. Draw the labelled drawings of the bipartite graphs K56 and K66. Answer the following for each graph by the built-in diagnostic test.
(a) How many edges has each graph?
(b) Is the graph eulerian? If it is, list an eulcrian walk.
(c) Is it hamiltonian? If it is, list a hamiltonian cycle.
3. Check the complete graphs K„, 2 =S n =S 7, and the bipartite graphs K,; (2    * < 5; 1 ^ j 6) for planarity, using a built-in diagnostic test.
Chapter 38
1. Rework Example 38.2 using a symbolic package for solving difference equations. Solve the mortgage difference equation
Q„ - (1 + I)Qm_, = -A, with / = 0.08 and Q0 = P = 50 000 (in £). Given that Q25 = 0, find A. List the outstanding debt Q„ each year m to the nearest £. Plot (a) the outstanding debt against years and (b) the annual interest repayments A - IQm against years.
2. Solve the following homogeneous difference equations:
(a) w„+2 - «„+i - L2z<„ = 0;
(b) un+1 + 2m„+1 + 2un = 0;
(c) a„+2 + 4w„+l + 4«„ = 0;
(d) uu+i + 3m„+2 + 3u„+l + u„ = 0, «o = 0, M, = 1, u2 = -1.
3. Solve the following inhomogeneous difference equations:
(a) m„+2 - u„+l — 12«„ = 2 + n + n1;
(b) un+1 - u„+l + 4un = 2";
(c) un+i + 3u„l2 + 3u„^ + u„ = n1, un = 0, ul = I, u2 = — I.
4. Devise a program to generate cobweb plots for the first-order difference equation
u„-i = ~kun + k for (a) k = j, (b) k = j, (c) fe= 1, with initial value m0 = | in each case (sec Example 38.3).
5. Display cobweb plots for the logistic difference equation
h„+i = auJX - u„)
for selected values of a. Some suggested values are:
(a) a= 2.8 to show a stable fixed point;
(b) OC= 3.4: find the period-2 solution;
(c) a = 3.5: find the period-4 solution;
(d) a= 3.7: chaotic output;
(e) a= 3.83: should be able to locate a stable period-3 solution.
6. Design a program to generate the period-doubling display shown in Fig. 38.11 for the logistic equation un^ = (Xu„{l — u„) for a increasing from (Z = 2.8to (1=4.
Chapter 39
1. (See Example 39.8.) A box contains 40 balls of which 7 are red, 12 are white, and 21 are black.
In each of the cases n = 2, 3, 4, 5, 6, 7, n balls are drawn at random from the box without replacement. What is the total number of rc-ball selections which can be made? What is the probability that there are n{n = 2, 3, 4, 5, 6, 7) balls of the same colour? Show the probabilities graphically in a bar chart.
Chapter 40
1. List the probabilities of the binomial distribution for n = 12 and p = 0.7. Check that their sum is 1. Plot this discrete distribution as a bar chart.
2. Plot graphs of the probability density function (pdf) and the cumulative distribution function (cdf) for the standardized normal distribution N(0,1).
3. Model a sequence of n Bernoulli trials with success/failure equally likely, in which the number of successes is recorded. You could try n — 50 run 500 times and count the number of successes i for £= 0,1,2,... , n. This should approximate to the binomial distribution nC:p'q"~'. Plot this distribution and compare
it with the simulation.
Chapter 41
1. Devise a program to draw comparative box plots for the examination data given in Problem 41.2.
2. Produce a histogram and frequency polygon for the pipe length data given in the table accompanying Problem 41.4.
3. Some randomized points (x„ >',) are generated by the Mathematica command
Table[{x+0.2*Random[],x+2+ 1. 2*Random[]},{x,0,6,0.5}].
Find the regression lines of y on x, and of x on y, for the data. Plot the data and both regression lines. Also find the mass centre of the data, and add this point to the graph. Where does the mass centre lie in relation to the regression lines?
4. Two dice are rolled and the average scores recorded. Compute the probabilities of the possible average scores, and plot them in a bar chart. Repeat the program for four and six dice. Plot bar charts in each case to illustrate the development normal distribution predicted by the central limit theorem.
1
Self-tests: Selected answers
Chapter 1
1.1 2<x=£3.
1.2 AB = BC = 4l6, AC = V52; ABC is a right angle.
1.3 The circles intersect at the points (0,1) and (2, -1).
1.4 The graph i s t
0 |*|>1.
1.5 cos ^71 = ^,   s.n^ = ^i.
1.6 sin(2arctan x) = 2x/(l + x2).
1.7 r = 1 + cos 0, which is a cardioid.
1.8 y=l/(l-x2e-").
1.9 Time T= (In 10)/L 1.11 (b) f(x)=-
(a-by-(x-a) (a-b)(x-b)2
(a-b)2(x-b)' 1
(c) f(x)=—^—
[x-aY (x-a)-
1.12 Sum to infinity is
3 2 (1-x)2 1-x'
1.13 (a) 5040; (b) 9990.
1.14 2[l + 2„C2,r2 + ,„C4x4+ - + ,„C„*21-
Chapter 2
2.1 Tangent: y = —2x + 2; normal: y = —4* + y; intersection point (^,y).
dV
2.2 -= 47lr2.
dr
2.3 ^ = 70(x'' + x>). dx
2.4 (a) 2; (b) 2; (c) 3.
2.5 d(cosh x)/dx = sinh x: d(sinh x)/dx = cosh x.
2.6 (2r)!/r!.
Chapter 3
3.1 dy/dx = eA(sin x + cos x).
dy l+x2-2x2lnx
3.2 — =-.
dx x(l+x-)2
3.3 dy/dx = 1728cl2v(l + Ue12')12.
3.4 dy/dx = kax In a.
3.5 dy/dx = 2xe>Jcos(e-,r2).
3.6 dy/dx = {x ^cos x[(2 + ln x) - 4x sin x In x].
3.7 dy/dx = (x-3)/[3x(l +2y2)]. At (1, 1), dy/dx = -§
3.8 dy/dx = 2/(1 -tanrrx).
3.9 dy/dx = —(£>/«)cot J. Tangents with slope (-1) occur at (a2, b2)N(a2 + b2) and (-a2, -b2)N(a2 + b2).
Chapter 4
4.1 (a) /"'(x) = ex[eos(x2) - 2x sin(x2)];
(b) fix1) =e1\cas(x4) -2x2 sin(x4)l;
(c) df(x2)/dx) = 2xc"![cos(x4) - 2x2 sin(x4)j.
4.2 x = 1 is a maximum, and x = 2 is a minimum.
4.3 x = 0 is a minimum, and x = 1 is a point of inflection (using a slope test).
4.4 The area change is 5A = 8jtr5r = 5.027: the exact change is 5.152.
4.5 Solution is x = 0.7686 to four decimal places, requiring three steps.
Chapter 5
5.1 1 + 2x + 2x2 + fx' + fx4 + £x5.
5.2 Required accuracy needs terms as far as x5.
„=n 2fz!
5.3 1   '   Jx2 + 4jX3. 5.5 -2.
Chapter 6
6.1 (a)4 + 3i; (b) i; (c) 2i.
6.2 í = 1 + i, ž = 1 - i, z2 = 2i, ž2 = -2i, 2í = 2 + 2i, 2« = 2 + 2;, zž = 2.
CO
111
to z <
Q LU h-
O
LU _j LU
CO
CO I— CO LU I-
LU
CO
6.3 z = 2(cos f + i sin f), z = 2(cos f + i sin f), 2z = 4(cosf+ i sinf ),z2 = 2V7(cos 0+i sin 0), where cos 9=lNv, sin 0 = -V(3/7).
6.4 z10 = 32i.
6.5 cos 40= 8 cos40- 8 cos20+ 1.
6.6 g = 2«7ti,? = ln(2±V3)+2«jci, (n = 0,±l,±2,
6.7 S(0) = cos (cos 0) cosh(sin 0).
Chapter 7
7.1 In full the matrix is
-1   1 1
-2 4 -8 -3   9 -27
7.2 AB--
1 0 0 6
BA =
4 -1 -1 -2 -2 -1
7.3 2A + 3B, A2, AB + BA are symmetric: AB and BA arc not symmetric.
7.4 A4 = abcdlv so that A"1 = A'Kabcd).
Chapter 8
8.1 dctA = 2(fe- l)2;k = l.
8.2 D,={x~a)"-\x + na).
8.3 The adjoint and inverse arc given by "-3   -fe-2   -2fe + 2T
-4      1 6 -1    fc+1 -2k-
3 fe + 2 2fe-2
4 -1 -6
1 2fe + l,
adjA =
4k+5
The matrix is singular if k = -|. The product A adj(A) will always be zero for a singular matrix.
Chapter 9
9.1 AD = (29, 35); direction is 0.878... rads to x direction.
9.3 Relative speed = 86.02 lm/hr; direction is 35.5° E ofS.
9.4 Plane is x -1 = X- 2jU, y + 1 == X, z- 2 = 3/L+ fl.
9.5 The point of intersection is (-1, -2p/(l - p), (1 + p)/(l -p)); the locus is the straight line x — 0, v+ z= 1.
9.6 r = -(B1r.
Chapter 10
10.4 (b)-45°.
10.5 (b) (1/W2, 1/\2,0).
10.6 Angle is arccos (—|).
10.7 (b) Perpendicular distance are 1/Vl4, 4/Vl4.
3c — 7     v + I     Z+ 5
10.8 (b) Line is —;-*= —r =—, •
Chapter 11
11.1 |c| = V26. 11.4 (a)-7» + 6/+fe.
Chapter 12
12.1 Solution is x, = 2, x, = -2, x, =-3.
12.2 A-
12.3
2 -1 -5 -2
5 2 9 4
7 3 13 6
L-8 -3 -15 -7.
(a) If fl^—3/2, the system has the unique solution
x-(a + b)l{Z+2a), >•••••• (-3+ 2i)/(3 (•2a'n
z = (a + b)/(3 + 2a).
(b) If (i = -3/2 and b # 3/2, the system has no solutions.
(c) If a = —3/2 and b = 3/2, then the system has the set of solutions x = A, y = —1 + 2/., Z = X.
Chapter 13
13.1 Eigenvalues: -1,1 - V2,1 + V2. Eigenvectors (-1, 2,2)1, (-1 + (1/V2), lHl, 1 )T, -1   (1A'2),-l/\2,1)T.
13.2 k*±l.
13.3 Eigenvalues are -2,1, 3. The corresponding eigenvectors are
(-2,1,2)T,   (2,-1,1)T,   (3,1,2)T.
A possible matrix C is given by
-2    2 3
1 -1 1
2 1 2
13.4
A" = — 15
20 L-18
-8 -12
-20 -30
18 27
-15    6 10"
-15   15 10
15   -6 15.
(-1)"
15
8	4	8~	17.7	x„+l f
10	-5	10		B+U
12	6	12.	17.8	
				(cr+1)
13.5 The eigenvalues are -8, 3, 5, and the eigenvectors are (-1,-3, I)1, (-3,2,3)T, (1,0,1)T.
Chapter 14
14.1 * = §-f cos3f + 4*.
14.2 (a) }e3v - cos 2x + C; (b) -3x_1 + C; (c)4 1n|x| + C.
14.3 (a) -je xl; (b) cos x csi"\
14.4 Signed area = e — e~' — 2; geometrical area = e + e ' - 2.
Chapter 15
15.1 Approximate area = 7?; exact area = |.
15.2 (a) j(b3-a3); (b) (e*1 -e'2)/x.
15.3 -ijsin1sx + C.
15.4 rms[/"(r)l =a/\'2.
15.5 |.
15.6 1/(1 + fc2).
15.7 sinh3x cosh2* is an odd function; cos3/ is odd about t - jJt.
15.8 l(x) ~ 2xex" cos{x1) - e* cos x.
Chapter 16
16.1 Volume = ffjt.
16.2 Area = jjt.
16.3 y = 2|;,y V(l - y)dy/Jl,(l - x1) dx.
Chapter 17
17.1 {x - ^ sm(6x + 8) + C.
17.2 -jcos(x2) + C.
17.3 h = -\ cos4x + C, 1, = —^cos4(3x + 2) + C.
17.4 Il = \n2,l,=\{\n2)1.
17.5 (a) 4-2 In 3; (b) k.
17.6 f(x + 3)-l+^ln|x+1|+f ln|x + 3| + C. 1
In .
, (n*-1);±(ln x)2, (n = -l).
n + 1,
■[2 - 2{a+ 1) In x + (a + l)2(ln x)2].
Chapter 18
18.1 x = eIOf-20.
18.2 (a) x = Ae' + Be"; (b) x = (A + Bt)c'.
18.3 x = e2' (A cos 3r + B sin 3ř).
Chapter 19
19.1 (a) x = f e"2'; (b) x = § cos 2i -{ sin St; (c).x-ř5 + ř'-|/-^.
19.2 The complex solution is x = — e~H"/(2 + i).
(a) x = e"'(—i cos t — \ sin ();
(b) x = e~'(j cos t — j sin ř).
19.3 A particular solution is x = -4i e~f.
19.4 x = f(-e < + e") -Tic"'.
19.5 x = —ý(cos t - sin i)c"'-cos' + C er"**.
Chapter 20
20.1 x(L) =2cos(3t + jK).
20.2 The amplitude of the superimposed waves is C cos j(0, - <pz). Cancellation occurs if 0, - 0, = 7t.
20.3 x={A+Bt)e-kl.
20.4 The resonant phase occurs at the polar angle given by {2k1, -2hi(a>l - 2k1)).
20.5 Nodes occur at z = [{In + 1)K + (0, - 02)]/2fe.
Chapter 21
21.1 X = 2eH
21.2 X = -0.1319 -0.00141Í.
21.3 p{t) = V(14 + 4V2) cos(5r + 0) where 0is the polar angle of (2^2,2 + V2).
Chapter 22
22.1 The isoclines are given by the hyperbolas x2 - y1 = constant.
22.2 General solution is x2(x2- 2y2) = constant.
22.3 General solution is x2y + xy1 + sin xy = constant.     Chapter 28
CO
cc
UJ
co <
Q LU H Ü LU _l LU CO
CO I— CO LU
LU
CO
22.4 General solution is xy1 = C(y- x)1, where C is a constant.
Chapter 23
23.1 The origin is the only equilibrium point. The equation of the phase paths is y2 = \x4 + C, where C is a constant.
23.2 For c < 0, the origin is a saddle; for 0 < c<\, the origin is a stable node; and for c > \ the origin is a spiral.
23.3 Equilibrium points are at (0, 0), (1, 1), (-1, -1), (1,-1), (-1,1). Solutions arex = +l,y = ±l.
23.4 The origin is a centre, the points (1, 1), (-1,-1), (1,-1), (—1, 1) are all saddle points.
23.5 Since r > 0 for r ^ 1, the limit cycle is stable.
Chapter 24
24.1 (a)2/(s2 + 4).
24.2 (b)2(l-e-M/(2s + 1).
24.3 L{ti£k•} =6/(s + k)4.
24.5 (s2-2s-6)X(s)-2s-3.
24.6 x(t) = -e2! + 2c3'.
24.8 £{(e-'-l)/t} =ln[s/(s + l)J.
Chapter 25
25.1 i(t) = (K/L) cos(WLC).
25.2 x(t) =-e' + 2e2'.
Chapter 26
26.1 The Fourier coefficients are a0 = 87l2/3, a„ = 4/n1, bn = -4n/n, (« = 1,2, ... ).
26.2 iTt
l-n-2
26.3 Sine series is > -sin 2ni.
^Jl(4«2-1)
Chapter 27
27.1 2/(1 +4n2f2)
28.1 (a) 3/73x = -2y cos(xy) sin(xy), (ifidy = —2x cos(xy) sin(xy);
(b) 9/7dx = -2* sin(x2 - y2), 3f% = 2y sin(x2 - y2);
(c) dfldx = [xy)'[i + ln(xy)], dfldy = x2(xy)rf;
28.3 Tangent planes are given by ±x + y — z — 2. The tangent planes intersect the x, y plane in
a square.
28.4 For maximum volume a = 2a/[A/(3a/3)J.
28.5 a-
6[2l«y„-(M + 1)Iy„l
N(N2-1)
, 2[-3l«y„+(2N+l)IyJ
b =-,
N(N-l)
where all summations are from 1 to N. 28.6 K(a) =37t/(16a5).
Chapter 29
29.1 At (3, 4), 8z = j8x +joy. The approximate change is -0.02.
29.2 Percentage increase in volume is approximately 9%.
29.3 In terms of x, the rate can be expressed as dz Ts' fori)--
- = V2(x-2)e
5 1.
29.4 dy/dx = -(x + y)l(x + 4y). The maximum occurs at (-2A/3, 2/V3) and the minimum at (2/V3,-2/V3).
29.5 The direction of the normal is (f+iVl3,f+|Vl3).
29.6 d/7ds = 2V5.
Chapter 30
30.1 dzldt = —.3 sin t(sin2t - 3 cos2f). Stationary at r = 0,ijt,§7l, jc, J7l,§71.
30.2 Stationary points arc at (1, 1), (-1,-1), (1,-1)
H,l)-
30.3 The families curves are confocal ellipses and hyperbolas.
Chapter 31
31.1 Maximum error = 0.261 units for an area A = 1.5 units.
31.3 The point is (f,f,j).
31.4 The tangent plane is 5x + 2y + 3z = 10.
31.5 The directional derivative (-2, -2, 1).
31.6 Restricted stationary values occur at (1,1, 2), (-1,-1,-2), (1,-2,-1), (-2,1,-1).
31.7 The envelope is the parabola y1 = x + \.
Chapter 32
32.1 / = ./ = 28.
32.2 l = {n.
32.3 Volume = 1.52/3.
32.4 J = e-'.
32.5 The moment of inertia is |M(a2 + b1), where M is the mass of the plate.
32.6 The volume is V = |jt.
32.7 Both areas = --.
Chapter 33
33.2 9/10.
Chapter 34
34.1 The field lines arc ellipses being the intersection of circular cylinders and inclined planes.
34.3 Surface area is |(5V5-1).
34.4 Volume = \Ah; volume of tetrahedron = -j^a1; volume of octahedron = jjrN2.
34.5 curl F = {x — 2yz)i — yzj - xk; curl G = (2y - l)i - 2xj-k.
Chapter 35
35.1 (a) 5, = {1,2, 3, 4, 5, 6, 7, 8}; (b}S2={ii,i,f,|,l,f}.
35.2 A U B= {x\xe Rand-1 S x^ 2, x = 3x = 4}; A n B= {1,2}.
35.3 (a) Same as Fig. 35.9b; (b) same as Fig. 35.9d; (c) elements which are not only in A or B or C.
Chapter 36
I
36.1 The output is a ''' b		
a b	a * b	
0 0	1	
0 1	1	
0 0	1	
1 1	0	
36.2 f=(a	*b)®	c. The
a b	c	f
0 0	0	0
0 1	0	1
0 1	1	0
1 0	0	1
1 0	0	0
1 0	1	1
1 1	0	0
1 1	1	0
36.3 f={a	*£)©	(a '■ b)
Self-test 36.1 have the same truth table.
Chapter 37
37.1 (a) {1,2,2,2,3}; (b) {2, 2, 3, 3}; (c) {1,1,1,2, 3};(d) {3,3,3,3,3,3}; (e) {4,4,4,4,4}.
37.2 21 arc connected of which three are regular with degrees 0, 2 and 4.
37.3 (b) (ii) A spanning rrcc could be the graph with edges [ba, bf, bg, be, ge, cd}.
37.4 ad(b + c)+eh(g + f).
37.5 By Euler's theorem: (a) the dodecahedron has 20 vertices; (b) the icosahedron has 30 edges.
Chapter 38
38.1 At 6.5% the repayment is £8198.15; at 7% the repayment is £8526.64.
38.2 The fixed point is ({(^3 - 1), |(V3 - 1)). The iteration gives u, = 0.460, w, = 0.288, m4 = 0.417, m5 = 0.326 to 3 decimal places, which indicates stabilitv.
m
m
u>
H
CO
CO
m
r-
m O
H
m o
CO
m J}
CO
38.3 un = (A + Bn + \n2)2".
Chapter 39
39.1 P(A1)={;(i)P(A2)=f;(ii)P(A,)=^
39.2 P(A U B U C) = ?{A) + P(B) + P(C) -P(Sfl C)
- P(c n A) - P(A n b) + P(A n b n Q.
39.3 The probability of six red cards is 0.0113.
39.4 Component is faulty with probability 0.942.
39.5 (a) 0.000125; (b) 0.1354; (c) 0.1426; (d) 0.1425.
39.6 -jy, the same as the probability for the second drawn component.
Chapter 40
40.1 The probability that the sum 7 occurs at throw i isp; = |(f)'-M' = 1,2,..4.
40.2 The probability p, that i individuals arc over-height is given by
40.3 Expected value is 2.8, and the variance is 1.82.
40.5 With n = 20, X= 2 the distributions are compared in the following table:
i	0	1	2	3	4	5	6
Binomial Poisson	0.122 0.13.5	0.270 0.271	0.285 0.271	0.190 0.180	0.90 0.090	0.0.32 0.036	0.009 0.012
40.6 /= a/{a[3+ 1); the probabilities are (a) yt0:
(b)
a
Chapter 41
41.1 The medians and quartilcs arc as follows:
	1 st quartile	median	3rd quartile	mean
Paper 1	41.5	47	56	47.7
Paper 2	47.5	58	63.5	54.9
Paper 3	43.5	50	59.5	52.5
Paper 4	45	49	65	54.3
41.2 ml = 1966; m2 = 2034. m, as n —> °°.
behaves like V«
Answers to selected problems
Full solutions of these end-of-chapter problems can be found at the website: www.oxfordtxtbooks.co.uk/orc/jordan_smith4e
Chapter 1
1.2 (a)y = -2x + 3; (b)y=l; (e)y = jx-±. Intersections are A : (2, 1), B : (f, {), C : (1, 1).
AB = iVl3,   AC = 1,   BC = |V5.
1.3 (b) Slope = j. Intersection with axes at (2, 0),
(0,-4).
1.4 (b) (y + 2)/(x + l) = -2, soy = -2x-4. (d) (y - 2)/{x - 1) = 3, so y = ix - 1.
1.7 (b) Centre (1,0), radius 2. (d) Centre (\, -\), radius fVll.
1.9 (b) x = -f ± jVl4, y = -} ± }Vl4.
1.14 (b)l. (d)-l/V2. (f)-V3/2.
1.16 (b)cosx; (d) -cos x.
1.17 (b) 2 cos \(x + y) sin \(x - y).
1.18 In the following, n represents any integer: (b)|jt + «7t; (A)\ + \n; (i)2n.
1.19 (b) amp. = 1.5; ang. freq. = 0.2; period = 31.41; phase = -0.48.
1.20 (b){x-\; (d)arcsin,-x,0«x=S2. (f) arccos(arcsin x), 0 =£ x =£ sin 1.
(h) -| + (1 + 4xf, x 3 -|.
1.22 (b)}e2: (d)}ln{, orIn 3; (f) 2; (h) +V2; (1) Hint: write sinh 2x = -j(e2* - c~2') and obtain a quadratic equation for e2*. x = -j In(4 + Vl7).
1.26 Hint: x = tanh y = (e* - £->)/(ey + <r-v). Form an equation for c-v and solve it.
1.28 5cos(fi)*-0.927).
1.29 C = 2, a= 1.386, f{2) = 1/8.
1.30 Tidal period = 12.57 h. It floats for 9.20 h. Hint: it floats when sin 0.5* 3= -0.666. Sketch
y = sin 0.5* and y = 0.666 and find the intersections.
1.33 The vertex is (-4,7).
1.36 (b)2/(x + 2)-l/(x + l). (d) l/2x-l/(x + 1) + l/2(x + 2).
(f) 1/4x-l/4(x + 2)-1/2(x + 2)2. (h) l/2(x-3) + l/2(x + l).
1.37 (b) l/[2(x-l)] + l/[2(x2+l)]-x/|2(x2+1)].
1.38 (b)x-3-l/(x + 1) + 8/(x + 2).
1.39 (b) 1 + 1/2+ 1/5 + 1/10 + 1/17.
6
1.40 (b)£(jr = (i)2 + (^ +- + (D6
= (i)2a + } + ■•• + (|)4].
Now (1.31) gives the sum in the brackets. Finally we obtain 121/729. (c) -341/1024.
1.44 (c) 1/99; (e) 30/11.
1.45 (b) 10/9; (d)2/3. 1.47 (b)(i)2mm!.
1.49 (c)256; (d) 20; (f) 59.
1.50 (a) 72; (b) 360.
1.51 (b)24; (d)164.
1.54 (a) 2880; (b) 720.
1.55 (a) 120; (b) 720; (c) 220; (d) 1000.
Chapter 2
2.1 (b)0.5; (e)2; (g) 1.
2.2 (c) 6; (e) -±; (g) -4.
2.3 (c)-1/x2; (f) 4x.
2.4 (c)-8.
2.5 (c) 32, -32.
2.8 {C)dE/dT = 4kT}.
2.9 (b)7x6-18x5+l.
2.11 Use the formula fortan(A-B) in Appendix B(b).
2.12 (b)|; (d)l; (g) 2; (i) 71/180 = 0.0175.
2.15 (a) 2 cos x+ 3 sin x.
2.16 (b)y = 24x-39; (d)y = c"'x.
2.17 (b)6x-2,6,0.
2.20 y = (-x + x„ + 2a2x30)/(2ax0).
co
LU _l
m o cc a
Q LU I-
o
LU _l
LU CO
O t— CO
cc
LU
CO
Chapter 3
3.1 (b) x cos x + sin x; (f) 2x In x + x.
3.2 (b) 1/(1 + x)2; (f) (x2 - 2x sin x cos x)/x4 cos2x. (m) nx"'1.
3.3 (d) f*S + g*Lt g#L + 2^ii + f*L,
dx      dx     dx2      dx dx dx2
dY, , dY dg     df d2g d3g
g--h 3---h 3---h f--.
dxJ      dx2 dx      dx dx2 dx
3.4 (b)-2 cos x sin x; (e) 2 sin x/cos'x; (j) I2x2(xj + 1)3; (n)-3e3-.
3.5 (f) $x-i; (i) -Ix"1.
3.6 (f) e~'(cos t - sin t); (k) 2 sin x(cos x - sin x)/x\
3.9 (c) (-2x sin x2)/cos x2. The original function only has a meaning when cos x2 > 0.
3.10 (b) e'(cos t + l cos t- t sin t).
3.11 (b) dy/dx = -yL'lxL-. This can be written in other ways; for example, put y1 = 1 - x1 from the equation of the curve.
3.15 (b)-5,
3.16 (b) dy/dx = +x/[2V(1 - (x/2)2)].
Chapter 4
4.1 (b)2/2; (c)4t3.
4.2 (c) x = e"'(min); (g)x = 0(min); (i) x = -1/V3 (min),x = l/V3 (max);
(t) Points of inflection at x = «71; maxima at x = (2n + \)1Z; minima at (In - \)n.
4.5 If base = x and rectangle height = y, then A — xy + jKx2 (constant), and P = (1 + \n)x + 2y. Substitute for y from the formula for A to express P in terms of x only. The minimum of P is reached whenx = [2A/(l + i7t)]'.
4.10 (b) 8y « -0.2 (exact value -0.227...). (d) 8y = -0.4 (exact value -0.5).
4.11 (a) 5v =-0.11; (d)5A = -0.08.
Chapter 5
5.1 (b) (1 + x)'' <= I + ±x - jx2 + -ijxJ.For 2 decimal places, we need I ^x3 \ < 0.005, or -0.43 < x < 0.43. (d) To four terms,
sin 2x » 2x - 1.333x3 + 0.267x5 - 0.025x7,
where (for this context) the coefficients are rounded to 3 decimal places. For two-decimal accuracy, we need -0.79 < x < 0.79.
5.3 (b) The terms in the expansion of sin x are of size jx|2"-V(2n - 1)! with n = 1, 2,.... We need to choose n so that this is less than 0.00005 when x = ±2. The first value within the limits is « = 7. The polynomial is
Uli + J-v13
1! 13! '
5.4 (b)-jTt-*.
5.5 (b)T + |x + {x2 + ---,-2<x<2. (h) 1 - jrx + jrx2----, valid for all x.
5.6 (b) 1 + \x - {x2.
5.7 (b) tan x * (x- fx3 + -^x5)(\ - W + irx4) 1
« x + \xs + £x5.
5.8 (d) ln(l+x + x2) =ln[x2(l + l/x+1/x2)]
= 21nx + ln(l + l/x+l/x2).
Then treat 1/x + 1/x2 as the small variable.
5.11 (b) Suppose that the first nonzero derivative is the N th : f[K'[c) # 0. Consider whether N is even or odd, and whether f iN,(c) is positive or negative.
5.17 (c)V + e-*).
5.18 (c)§.
Chapter 6
6.1 (b)3±i.
6.3 (b)3-5i; (d)9 + 3i; (f)l + 6i.
6.5 (d)
6.6 (a) -4i; (c) -j + }i.
6.7 (a) 1-i; (c)-2i.
6.8 (b) 16.233-0.167Í; (d) 88.669.
6.9 (b) |z2| = 8;Argz2 = -fiC. (d) \zt\ =3; Argz4 = 7t.
6.10 (b)y = 2; (d) the parabola, y1 = 4x; (f)y = x(x=*0).
6.11 (a) ^2e>; (d)14c       (gJeV; (j) ^2t^'.
6.16 (a)2«Jti (« = 0,+l,+2,...); (c) (2« + l)jli.
6.18 (a)cos(ln2)+isin(hi2).
6.23 (a) x2- y2 + 2xyi.
(d) cos x cosh y - i sin x sinh y.
6.28 2 + i,2-i,-l-i,-l + i.
6.29 (b) e2a"»cos(2 sin 0).
Chapter 7
7.2 x = -2,y = l 7.6 BA
-10 -5 20 10
7.7 A2 + C2 =
-5 6 16 -8   11 2
-6  -6 -7
7.11 A2""1.
7.16 x = -17,y = -2,2=8.
Chapter 8
8.1 (c)l; (e) - I.
8.4 (b) 1728; (d) -8132.
8.6 (b-c)(c-a)(a-b){a + b + c).
8.14 x = a, b,c,—a — b- c.
8.16 det(AB) = -36, A"'
7 1 -2 0
1   1 -1
Chapter 9
9.1 (a)PQ = (5,-3),QP={-5,3).
9.2 (f) Length = 5,0 = 126.9°.
9.3 (b)
3 3*3 2' 2
9.4 BE = (0, -4); BE = 4; bearing south.
9.5 (c)V6.
9.7 (b) 2a = (6,4,6), ib = (3,3,6), 2a - 3b = (3,1, 0).
9.10 (a) (3,3,-6). (b) (X+ 2)2 +(Y-1)2 +(Z + 3)2= 1.
9.16 Speed 10*^2; direction towards north east. (Hint: use vm = t>w — vc in components, with t>N = (k, v).)
9.22 (b) fa + \b; (c) f« - ifc.
9.23 (a) (a + Xb)l(\-X). (b) («-Afc)/(l + A). (c) The point is on the extension of AB in the direction of AB.
9.26 (a))> + z=l. (b) 3x-2y-z = 0.
9.27 V2.
9.28 (a)±(3/V34,4/V34,3V34). {b)±{j,jr,±).
9.29 (a) -3* + 2/+ 4fe. Length \29.
1 f a ^1
9.36 r = —--1--. (Hint: draw a diagram
2\J<*I    1^1/ involving4and
9.37 The minimum separation occurs when t = 12 j s.
Chapter 10
10.1 (a) 10. (e) zero.
10.3 If your diagram is a parallelogram ABCD, the theorem obtained is AC2 + BD1 = 2(AB2 + AD2). If you use the triangle rule the result gives the median of a triangle in terms of the sides.
10.5 (a) 6. (b)-5.
10.6 (a) 35.3°.
10.8 54.7°.
10.9 33x2 + 13r - 95z2 + 48xy - 144yz + 96z.%- = 0.
10.10 32.5°, 78.9°, 68.6°. 10.12 F=      a + ffb + f-c.
10.16 a = -f 0 = f, y =
10.17 x = 0,y = 0,z = l.
10.18
(a) (2^2,0). Cb)X-^   + Y + ^ =1
1
10.19 (c) (/, m, n) = (}, -f,-j). 10.21 {a)±(A ± 10.26 (a) 19.1°.
10.30 (a) P1k-y + z = 4, P2 is 2x-2y + z = 5. (b) 45°. (c) 2*2. (d) and
(e). The line L is given by r = A(l, 4, -4). Show that intersection with P1 and P2 occurs when A = —j.
10.34 Begin by finding any two points on the line of intersection. (The resulting form is not unique.)
Chapter 11
11.1 (a) (4,7,5). (d)-9. (h) (-24,3, 15).
11.9 Hint: the determinant is equal to a • (b x c), where QA = a, etc.
11.12 X:
-hy-
■4,z =
11.13 (c) A = jU = —j, f = —f. L, meets L, at (-1,0,-*) and L2 at (j,-f",-T).
11.15 (a)(f,f,-4).(b)(f,f,-f).(c)fi. (Note: the wraY vector in the direction o(i-2j-2k is±(*-2/-2fe).)
11.16 (a) -6, (b)6. (c)0. (d) 0. (e)-2\3.
Chapter 12
12.1 (c) xl = l,x2 = -l,xi = -5. (e) x, = 2, x2 = —1, .Vj ~2,x4 = 2.
CO
LU _l
CQ
o rr
0.
Q
LU h-O LU —I LU CO
CO DC LU
co
12.7 *! =		= 40,		xl=	88, x	
			-	5	0 -	5
12.9 (b)		25		-6	10	1
				7	5	3
	1	0		0	0	o"
	-1	1		0	0	0
(e)	0	-1		1	0	0
	0	0		-1	1	0
	0	0		0	-1	1
-59.
12.12 The shadow on the z plane has vertices at the points (-1,0,0), (-1,-2,0), (1,0,0).
12.16 Non-trivial solutions if k = 1, -1, 4.
12.18 Non-trivial solutions if k = -6, -1, 3, 4.
12.22 a"| = 1.398, x2 = 1.090, x, = -0.2844, x4 = -0.3697.
Chapter 13
13.1 (b) Eigenvalues 4, 9. Eigenvectors (e) Eigenvalues 3 - 4^2, 3 + 4\2. Eigenvectors
—3		l"
2		1
-1 - 2V2~		~-i + 242
7		7
13.4 (c) Eigenvalues (-2, 2,3). Eigenvectors
0		1		0		" I
-1	>	0		2	15.4 (b)	(x1
2		0		1	j	-1
13.7 a = -2 and a = —\. 13.12 The matrix C is given by -1 -
C-
7
-1 -1 -1 0
13.16 limA"
1 1 1 1 1 1 1   1 I
13.22 Eigenvalues are 0,4, 4, 12.
13.26 A3" = I„ A3"+1 = A, A"'*2 = A2.
Chapter 14
14.1 (a) {x6 + C; fx5 + C; \x* + C; }x3 + C;
ix^C.^ix + C; C.
(g) e> + C;-c- + C;
je2* + C; -2e$* + C; -f e"2* + C.
(k)x + lnx + C ((x + l)/x = l + x '); 2x - 2xT + C; In | x | - 2x~' - \x 2 + C.
14.2 (b)
-}(1 - xf + C; -f (8 - 3x)^ + C; {(1 - xp + C.
14.3 (b) -In [1 - x| + C; -} ln|4 - 5x| + C.
14.4 (c) fx + } sin 2x + 4j sin 4x + C.
14.5 x2 e* - 2x cv + 2 ex + C.
14.6 (a) 2; (h)-ln2.
14.7 (c)4-x23=0if-l=£x=£2, and
4 - x'' =S 0 if 2 =s x =S 3. The geometrical area is
|E(x)|2, + |F(x)||, where F(x) = 4x — fx3.
14.8 (a) At + B; (b) \t' + At + ft.
Chapter 15
.«=1 r i
15.1 (b) lim £ x5 8x =     xs dx = [jx6]1^ = 0.
15.2 (b)    (x + lp dx = |(x + 1)"; + C.
15.3 (c) ( dx = [x]2=2; (i) 4(2* -1). lx = [jx3-xjL1 =-f,
15.5 (b) I   tivdv = -2 [e '"]~ = -2(0 - 1) = 2.
15.6 (c) 2/JC; (h) 1
T
(l-e-')dt = T + c-T -1. (T + e T - 1) = 1 + T"1 e"'' - T"' -» 1
as T —> °°.
15.7 The integrands are (a) even; (b) odd; (c) odd; (d) odd.
15.9 (b) The exact result is Vjc/2.
15.10 (e) \(x + 1)~T sin (x + 1) - \x 1 sin x.
15.11 (b) x =S -1: \ (constant); -1« x « 1: ^x2; x 3= 1: \ (constant)
15.14 6.
Chapter 16
16.1 5.3 x10~3.
16.2 (20 - 10() it = -20, x(4) = -17.
16.3 (b)jJU; (g)7t. 16.5 (a)|jwfc2.
16.6 f =
7tx2dy =
Jt(2y)2dy = 287t/3.
16.7 Put x = 0 at A; moment =    wxdx = 4w*£2.
. (i
16.8 1.18.
16.9 0.015 g.
16.12 A sketch shows that x(x - 1) ;- -x if 0 x =S 2. Therefore the area is
lim Y [x(x - 1) - {-x)\5x =    x2dx = f. 6»-»0 '—'
*-0 Jl)
16.13 f.
16.14 (b)Jt.
16.15 In a plane perpendicular to the end, y is downward and x is horizontal; the origin is at the top. Area elements are horizontal strips of width 8y in the end face. Force = jpglJP. Moment = ^pgJ,H:\
16.16 Distance of centre of mass from vertex isJH.
16.17 -jj<7a3b(<T= mass per unit area).
16.18 (a) jOBH'; (b) ^OHB3, where cris mass per unit area.
16.23 8a.
Chapter 17
17.1 (c) - j c"3* + C; (f) -£(3 - Zxf + C; (j) (2x - 3)i + C; (n) i\n\2x + 3| + C;
(o) ]n|l-x| + l/(l-x') + C.
17.2 (b) -f cos i(3r - 1) + C; (c) -|(-i)j + C;
17.3 (d) 4 sin(x2 + 3) + C. (j) } ln(l + x1) + C.
17.4 (c) j sin32x + C. (g) Put cor 2x = cos 2x/sin 2x, then u = sin 2x, giving |ln|sin 2x| + C.
(j) |cos3x - cos x + C.
17.5 (b) 205/32; (e)-In 2; (h)4ln2; (k) zero; (n) {21(0) cos (j).
17.6 (b) \k;   (d) 471 + 4;   (f) fit.
17.7 (e) tan x - x + C; (f) -x ' - arctan(x ') + C; (k) ijarcsinx + x(l - x1)1-] + C.
17.8 (b) j In x/(x + 2)| + C. (d) ln|x + l|-4ln|2x + 1| + C. (f)ln:xj-41n(x2+l) + C.
(i) 4ln[(l + sinx)/(l-sinx)l + C.
17.9 (b) jln(x2 - 2x + 3) + C. (e) ln(ex + e"1) + C. (f) 2ln(x4 +1) + C.
17.10 (b) \xe3x - 4e3r + C.
(f) 2x sin 4x + 4 cos 4x + C.
(i) 4x2 In x - 4x2 + C. (j) x"+1 [In x - II (n + \)]l{n + 1). (k) Hint: bring together the two terms J (In x/x) dx.
17.11 (a) Hint: there are two stages required; see Example 15.20.
17.12 Hint: the same integral occurs on both sides but with a different factor.
17.13 (b) zero; (d) {; (h) K.
17.15 F(0) = 4-ti, F(l) = I, F(4) = £jt, F(5) = £.
17.16 (a) 2 (In 2)3 - 6 (In 2)2 + 12 In 2 - 6.
(b) F(0) = 2, F(1) = Jt, F(4) = 7t4 + 127t2 + 48, F(5)=7t5 + 207t! + 12071.
17.23 (c) {alb) arctan \{a tan x)lb] + C; (d) In(tan4r) + C;
(g) ln secx + tanx| + C; (j) ln[(l + V5)/2]; (k)8;6\3 + 1VI5.
17.25 Coordinates of centroid: ()/;, 0).
Chapter 18
18.2 (b)a-=Ae4'; (e)x = Aci'; {i)x=Ac'.
18.3 (b) x = c1^; (d)x = 10e-;,+li.
18.4 l{t) = ln e~Rt". I reduces to a fraction 1/n of itself in any interval of length (L/R) ln n.
18.5 (a) A{1) = C e~k' (C arbitrary).
(b) The half-life 1 = — In 2 years. The information
implies that c"** = 1-0.175 = 0.825, sok = 0.0096. Therefore T = 72 years.
18.6 If N(*) is the number, then 8N = 20(jN) 5f so the equation is dN/dt= 10N. In the second experiment there is an average death-rate of 1 per rabbit per year, so AN I At = 9N.
18.7 (b) Ae' + B e~~. (e) A e"2* + B e-"'!*. (1) A e ■" + Bt e-3'.
(n) A + Bt (this is an exception to (18.10)).
> Z to
m
to H O to m r-m O
H
m O
TJ 3d O
CO
I-m
to
00
lj _J co
o a
Q LU H O LU —J LU CO
o I-
00
cc
LU 00
18.9 (b) f(e'-c-2').
(d) The general solution is A e~' + Bx e~v, y = e(x - l)e~*.
18.10 (b) A cos 3* + B sin 3f.
(d) A cos u>0i + B sin fl)0t. (f) e'(A cos f+ B sin t). (i) e"?''(A cos \^2t + B sin jV2')-
18.11 (c) a cos (O0t + (b/fi)0) sin (buf.
18.12 0 = a cos(g//)*f.
18.13 The initial angular velocity dfl/df is iV/;
0 = -
n| y I I
18.14 0= 0.0719 c"0033' sin 0.696^. 18.18 A = (Mg/P) e^-Hip_
Chapter 19
19.1 (b)-}t3-jf2-jt-f. (d) }e2'. (i) -£sin3£
(k)      cos 2* + -jj sin 2/.
19.2 (d) j(-6 cos t-3 sin t).
(f) -1§f(4cos2f+llsin2i).
(h) ^-e'(4cos2f + 7sin2f).
19.3 (b) -jtcos2t.
19.4 (b) \f-e-, (e) \te' sin*.
19.5 (c) Ae4' + Be"='-l-2 cos 2/.
(i) A cos x + B sin x + x1 - 1 + }e3\
19.6 (c) -{ + Ae'\
(g) (sin x — cos x - x cos x + A)/(x + 1). (1) (x + 1) ln|x + l| + 1 +A(x+1).
19.9 ll.-j minutes.
Chapter 20
20.1 (b)3cos(<at + Tc). (e) 3 cos(2f + |ic).
(h) 5 cos(2t + <j>), 0 = —arctan -j.
20.2 (c) x leads y by It.
20.3 (b) (i) 0.318 cycles/s. (ii) 0.316 cyclcs/s. (iii) About 3 cycles.
20.4 (b) C = V(4- V6), 0= arctan(l/(V6 - 1)), (-t7C<^<0).
20.7 The solutions are of exponenrial type.
20.8 jt = e-*-4e"*.
20.9 A e-fa + Bf e-".
20.10 (a) Period = 1.0508.
(b) Amplitude = 10/[(36 - m1)1 +<o2p, phase = -arctan[to/(36 - CO1) J.
(c) Resonance: (0=5.958.
Chapter 21
21.1 (b) -2e^ :2e "'in standard form).
21.2 (d) 2e"^';2cos(of-|7t). (i) eL,7i; cos (Q)t+ 1.97).
21.3 (b) 1 - e^"' = 1 + i = V2 e'"1.
21.4 (b) 1 - 3e"^i + e** = 1 + 4i = <f\7€*, where 0 = arctan 4 = 1.33.
21.6 (b)R + (0Li. (d) R/(l+wRCi). (i)R + icoL/(l-w2LC).
(k) ia)RL/\R(\ - «2LC) + iof.].
21.7 V=Zrand V=2.
(d) / = 2(l + i«RC)/R;|I] = 2(1 + <b2R2C2)'7R; arg/= arctan(c)RC).
21.8 (b) V,/V„ 2i); V,//, = {(5 - i).
Chapter 22
22.4 (b)2x2-/ = C. (g) y = x/(1+C.v).
(k) x = +2~4(C-f'pforr' <C.
(n) arctan y + arctan x =     Take the tangent of this
expression and use the formula for tan (A + B); we
find thaty = (x+ l)/(x-l).
22.6 (b) y = -^{x2 + C)2 for x2 + C> 0. y = 0 is also a solution, (d) Those parts of the curves y — sin(ln |x| + C) for which x and dy/dx have the same sign. Also y = +1 arc solutions.
22.7 (b) y3 - 3xy = C. (d) xy -y1- x1 = C. (f) j»3 + y - x1 = C. (h) y + cos y + sin x = C. (j) e'+-v + y-x = C.
22.8 (b) xy + ylx = C; (d) x/y + y-x = C;
(e) ylx - xly - 11 x = C; (f) x2/(2y2) + 1 /(xy) = C.
22.12 (b) x(l + 2y1/x1r = C. (d)x2-4y2 = Cy3.
Chapter 23
23.2 (b) y = Cx (this is not covered by (23.22)). (d) xy = C (a saddle).
23.4 (b) Saddle (i.e. unstable), m = }(-3 + \j\3).
(f) Stable spiral; directions arc clockwise round origin.
23.5 (b) Equilibrium points at (1, 1). (1, 1) is a stable spiral, anticlockwise about (1,1). (d) Equilibrium
points at (-1, 0), (0,0), (0,1); (0, 0) is a centre and :'•-•], 0), (1, 0) are saddle points.
Chapter 24
24.1 (b) 4/(5 + 1); (d)6/s3-l/s; (g) (3 -s)/(s2+ 1).
24.2 (b) l/s-2/(s + 2); (e) (3s-4)/(s2 + 4); (g) i[l/s-S/(s2 + 4)].
24.3 (b)l/(s + 2)2; (d) (5-2)/(s2-45 + 5); (i) (52-9)/(s2 + 9)2; (1)24/(5 + 1)'.
24.5 (b)l; (d) {i4; (g)|e^ (k)|e'+|e';
(0) 2cos2i-jsin2i; (s) \ e't1; (u) }(cos t - cos It).
24.6 (e) (2s2 + 35-2)X(5)- 105-9.
24.7 (b) 2 e' + e 2'; (e) 3 e"' cos It; (f) y = | e* + \ e * + { cos x.
24.8 (b) 3 - 3 cos t + sin t.
(e) -{ e~' + f e! - \t e* + }r2 c'.
(1) -le'-lc-' + fe21-^-2'.
24.9 (b) x = J + | e4' + |« e4<; y = -± + ± e4' + \t e4'.
24.10 (b) e'(jA + \B + f) + e"'(|A - JB + {) - 3, where A and B are arbitrary. This is the same
as C e' + D e"' - 3, where C and D are arbitrary.
24.13 e"2 e-2sf(5 + l)2- l]/[(s + 1 )2 + 1 ]2 = e-2 e-2' 5(5 + 2)/(s2 + 2s + 2)2.
24.14 (b) H(t) sin t-H(t- 1) cos(t- 1).
24.15 (b)(ic2' + {e-2'-i)HW, -(|e2'-" + |e-2<'-1'-|)H(«-l).
(d) \H(l)t sin r + ^-H(r - 7t)(r - 7t) sin(f - 7t).
Chapter 25
25.3 Hint for working: s2 + 2fes + ft)2 has real factors when k2 > (O2; so put s2 + 2fes + CO2 = (s - a) (s - /J), where a,ji=-k± (kl - a2)1. Then x(r) is given by
(a-j5) '[(a+k) e"'-(/3 + x-)e^']H(f) + /(a- /3)-'[e"<M»;i - e«M»']H(t -10),
where *• = 1 + Ik.
25.4 By proceeding as suggested, we obtain
u(x) = Ax + \Bx> + (Mg/6K)(x - \l'fU(x - \l).
The conditions at x = I give A = Mgl2l\6K, B = —Mg/2K. This problem could be solved by integrating the equation four times, and linking the solutions over [0, jl] and     1} by the condition that u(x), u'(x), u"(x) are continuous at x = but this is automatically secured in the Laplace-transform method.
25.5 (b)25/(6s2 + 5 + l).
25.6 (b) V2/V, = 3/(20s2+ 125 + 5); V2/7 = 3/(4s2 + 1).
25.7 (b) f; (f)l-cosf; (h) \(—t cos t + sin t); (j)n!m!t"+"'+1/(« + m+l)!.
25.8 (b) — /r(T)(e<""-I'-e-,0<'-T))dT.
2c0)o
25.9 (b)coshf.
25.19 (a) x(t) = 6(f) + 28(f- T) + 8(7 - 2T), X(5) = l + 2e"s' + e-2sT.
25.21 {a)z'+2z-2-z-3. (b) 1 - Z"1 + z"2 - ■■■ = «/(?+1). (c)2«/(2«-l). (d)z/(z2-l).
25.22 (a) Tzl(z- l)2.
25.23 (a) (z - l)/(z + 1), g(i) = {1, -2,2, -2, ,.. }.
25.27 (a) Unstable. Poles at z = ±2, giving growth |2"
and |(—1 )"2". (c) Stable. Poles at z = +{i, giving decay
1 1 1 --cos — Tin.
4 2" 2
Chapter 26
26.1 (b)a„ = 0,fe„ = -2(-l)"/».
(e) a„ = 0, /?„ = —[1 + (-1)" - 2 cos(inTt)].
7t«
26.2 (b)k„ = 0, ag (c) bn = Q,an
_ 2n2 3 ' 4(-l)"
-(-!)"(« = 1,2,...).
7t(4«2 -1)
26.3 (a) a0 = -jJt,
: 0, a,n. = ■
JC«Z
(« = 1,2,..,). n
26.5 Series sum is jTC. 26.8 F = 2.
40
26.10 a0 = 0,a„ = 0,fe„=—^-[l-(-!)"](» = 1,2, ... ).
26.16 (a) T-sin(2« - l)TCf.
~f (2n - l)7t
2    ^ 4 26.18--> -cos 2nd)t.
7i  t! K(4"2 -1)
1 4
26.23 (b)R(<) =
cosTCf -I— cos 3tk
32
2 Tt 1
+ — cos 57lf + ■
52
26.26 (b) £ —
26.30 4 + — ^ -e,2"""T. 2    2k it n
Chapter 27
27.1 XJf)=4nf!(l + 4n1f1); Xc(f) =2/(1 +4n2f2), 27.8 XW = 2|        cos 27r/jdt where
X(f) = 2 x[i)cos2nftdt.
Jo
27.11 (c) 2c sine c/xos 2nbcf.
27.12 (a) 4 smc-'-rf. (b) \ smc2\f canf.
27.17 {l/[a + i(27t/"+ /3)] + l/[a+ i(2ir/-/?)]}.
27.19 (b) l/(l + i2Tt/)2.
27.20 (b) The Fourier transform is sinc2( f) er'l<a+'"f <r>A[t-{a + b)~].
Chapter 28
28.3 (c) 4x-2y- 1; -6y- 2x - 1.
(f) y - 2; x - 1. (i) 2y/(x + y)2; -2*/f* + y)1. (k)        + y2p; y(x2 + y2p.
28.4 (c) — = g'ir) cos 0; —— = g'(r) sin 9.
dx ay
28.8 dlf/dx2, d'f/ay2, and r)2/7r)x 3y = d2f/dy dx are given in order: (b) 2, 4, 3. (d) 2y/x\ 0, -1/x2. (h) 108(3x - 4y)2,192(3x - 4y)2, -144(3x - 4y)2. (k) -r~3 + ?ix2r~\ -r'3 + 3y2r 5, 3xyr_i, where r = (x2 + y2)1.
28.10 (b) 2x + 2y-« = 4; one normal is (2, 2,-1). (d) 3x + 4y + Hz = 29; one normal is (-4, -2, -1).
28.11 78.9° or 101.1°.
28.12 (b) (1, -1), min; (d) («7C, mil); min if n and m odd, max if n and m even, otherwise saddle;
(h) (0,0) saddle; (1,1) minimum; (k) (0, 0), saddle.
28.14 (a)a = fo = c = 7; (b) a = b - c = 4.
28.15 The maximum is 9, attained at (2, ±1).
28.16 Minimum distance = \'2.
28.18 (b) Depth = 2~*V \ square base, side 2*V*. 28.23 Lowest point is z = ja at (0, a) and (a, a).
Chapter 29
29.1 (b) 8« = 0.0718... (exactly). The incremental approximation gives 8z = 0.0784. Error = 9.1 %.
29.3 (b)-8y(5» + 5y)/(l + 8y).
29.6 -5.7%.
29.7 1.67% reduction, approximately.
29.9 (b) -2v2; (d) zero (it is the same in all directions).
29.10 (b) -{; (e)        (j) 1.
29.12 (b) xlxia2 + yj/b2 = x\ la2 + y\lb2.
(f) axjx + h{y,x + xLy) + byy\ + g(x + xt) + f(y +y,) + c
= 0.
29.16 (b)x'
= constant; (d) e* + c] = constant.
29.17 (b) y2 - x2 = b2 - a2. 29.19 (b) 49.8° or 130.2°. (d) Hint: compare Problem 29.12f.
29.21 (b)(0,j); (d)(-4,l).
29.22 (b) (2,1)/V5.
29.23 (b) 0 = 0.
Chapter 30
30.2 (b) -4 sin t cost; (d) 2 sin(f2) + 4(2 cos(/2).
30.3 It is easiest to start by expressing the distance D in terms of polar coordinates (r, 9), (R, (j)) by using the cosine rule (Appendix B(f)). Then
d£ _        (Rf-rV) sin (0-0) dt       [R2 + r1 - 2Rr cos(0 -9)$' where 9=vtlr, <f> = VtlR.
30.4 (b) x — y = 3. (c) The coordinates of the nearest point on the given line are (j, y). Distance = iNS.
30.5 (b) (0, 0), (2, 0). (A suitable parametrization is x = 1 + cos t, y = sin t.)
(d) (+6A/5,±4/Y5). (A suitable parametrization would be x = 2/cos I, y = 2 tan t.)
30.8 (b) x = -2r9 sin 9+ r cos 0- 0V cos 0- 0r sin 9, y = 2r9 cos 0 + r sin 9 - 92r sin 0 + 9r cos 0.
30.9 (c) 3/73« = -2vLlu\ dfldv = 2vlu.
30.10 (b) 32/7dir = 12m2 - 2v2, d2fldu dv = -4uv, d2frdv' = -lui+\2i: .
30.11 It is easiest to put x2 - y2 in terms of uv. Finally, d2fidu2 = \6v2g"(4uv), d^-f/dv2 = I6u2g"(4uv), d2fidudv = 4g'{4uv) + Uiwg"(4m>).
Chapter 31
31.1 (b) 8f = -x(xl + y2)~^e-■' &x
— y(x2 + y2)"e~'8y — (x2 + y2)_Ie~' 5t.
(e) bf ~ 2(x1 - x,) 5x, - 2(xt - x2) 5x,
+ 2(y, - y2) h\ - 2(y, - y2) 8y2.
31.2 -0.07.
31.3 Tt is easiest to write 5(1 /R) = -SR/R2. Wc obtain 8R = 0.198 SR! + 0.018 8R2 + 0.334 5R,. The required 5R, is -0.108.
31.4 Put ax' - bx - c = f(a, b, c, x) and use (31.1).
31.5 (b) Hint: use logarithmic differentiation: Sw ~ -3 8x + 3 Sz and bw ~ 2(+0.6). What is the significance of the absence of a term in 8y?
31.8 (b) 28x + 48y - 65z = 0. For dz/dx, put 8y = 0: dz/dx = j. Similarly 3z/3y = |.
31.11 (b) (2,-3,5); (d) (3.y',0,9z2);
(f) (—x/r3, -ylr3, ~z'r3), where r= (x1 + y2 + z2)T.
31.12 (b) (0, ly, 2z). Unit vector = (0, y/(y2 + z2p, z/(y2+z2F).
31.13 (b) cos <p= ll/(3Vl4), so (p= 11.5° (i.e. the angle of intersection of smallest magnitude).
31.15 (b)s-(2x,-2y,-3).
31.16 (b) (Check that i as given is a unit vector.) df_ ds
31.17 (b)-2/-2fe.
: 7.51.
31.18 (b) (±1,0,0) and (±1, ■*-,£).
(d) x = y = z is a line of stationary points (excluding the origin), (e) x = y - z = ±1, V.3, A = ±4V3.
31.19 Stationary at (1,0,0), (-1,0, 1), (-1, 0,-1). 31.21 (b) (3,3,3); te)-(aN3,bN3,cN3); (g) (7,7,!). 31.26 (b)4.xv=l; (d)x2 + y2 = l.
Chapter 32
32.1 (b)c-2; (d) (d- c)(b -a); (i) -j; (m) {In 2.
32.2 (b) Zero. Refer to the signed-volume analogy (30.2b). (f) ln(27/16).
32.5 (b)
(d)
/"(x, y) dy dx.
f(x, y) dy dx.
fix, y) dx dy +
fix, y) dx dy.
32.6 (b) 4; (d) f; (h) £
32.7 (b) 1.
32.8 (b) J7t; (d) f; (f) f.
32.9 2a3(4 + 37t)/9.
32.10 (a) 2(u2 + v2); (b) 2; (c) 1/5; (d) -2 cosh v.
32.11 The value of the integral is 2(257 - 129V2)/5.
32.12 Area =1/12.
32.13 1/e.
32.14 Volume = 64/3.
32.15 1/4.
32.18 (a) VTC(|H-|a|).
Chapter 33
33.1 (b) 1.
33.2 (b) i; (d) f; (f) 0.
33.3 (b)Jt; (d)~Tt\
33.5 (b)2; (d)0; (g) 0.
33.6 (b)l; (d)-3; (f)f.
33.7 (b)0.
33.8 (b)-|.
33.9 Zero.
33.11 Putx = x{u,v) and y — yiu, v), where u
and u are the new coordinates. Then put
,     dx dx ax = —-du + —— av etc. du av
33.14 fit.
33.16 (b) Non-conservative.
Chapter 34
34.1 Ti[a3-(a-b}3]Ja.
34.2 (a) 1/84; (b) 1/24; (c) 13/384.
34.5 2V6-2sinh '.;\2i.
34.6 Scalar potential is eT1* + cos xy + zx + C.
34.7 Scale factors are ht = b2 = \'(k2 + v2), h3 = uv.
34.11 (b)divF = 2z.
34.12 (b) curl F = 2x1 + (x - 2y); + k.
34.14 (a) Srh (b) 0; (c) 3rr; (d) 0; (e) 0; (f) 12r. 34.18 0.
c
m
3D CO H
o
CO
m r-m O rl m O
13 3) O 00
r m
co
co
111 _l
OQ
O
DC
dl
Q UJ h-Ü LU -I 111 CO
CO
cc
UJ
CO
z <
Chapter 35
35.1 (c) -2, -1, 0,1,2,3, 4; (f) 1,4,9.
35.3 (c)AuB= {-4, -3, -2, -1, 1, 2, 3, 4}.
35.4 (b) A n B = {x\xe M* and -.5 =e x =S 2}. (d) A n ß = {1}.
35.5 (b) B\(A u C); (d) (B n D)\A.
35.6 (b)S,\(SuS2u---uS,.).
35.7 (b) [(A\A,)\B,JuB2.
35.10 A2 = {(1,1), (1,2), (2,1), (2,2)}. 35.12 (b)66.
Chapter 36
36.6 See table below.
36.10 (a) (a*fc)*(fc®c); (d) (a »fc © a*b){c*d).
36.15 (a) If a, represents the state of switch S,, etc., then the switching function is
(«i ©a2) © [(a, ©a4)*aj].
36.16 See the table below.
Solution for 36.6
Solution for 36.16
a		c	[a@b)*(a	®c)		a2	a3	f
0	0	0	1		0	0	0	0
0	0	1	0		0	0	1	1
0	1	0	0		0	1	0	1
0	1	1	0		0	1	1	0
1	0	0	1		1	0	0	1
1	0	1	1		1	0	1	0
1	1	0	1		1	1	0	0
1	1	1	1		1	1	1	1
Chapter 37
37.3 Twenty are planar.
37.4 Five are connected.
37.8 Six not including reversed order.
37.13 There are three different paths between a and e.
37.14 Five vertices.
Of.13   li —     21 U'  2 —     2K0'J3—     21 0' 4 21MI>
. _         1  ■     ■ _        4 -     - _ 3 ■
/5 — ^i^o, *6 — TT*0> *7 — — y\y
37.17 The transfer function is
q__j)g1g2g3_
1 - g,h, + G1G,G3H2'
37.19 (a) The transfer function is PGfi2G3
37.20 (a)
(d)
1 + G2H, - Gfifi^H, ' G,G,
a-GÄH^a + GjH,)' +
1 + G,G,H2     1 - Hj
37.24 SAFT, length 12.
37.25 2 ties.
37.27 (b) Framework is overbraced.
37.28 Two ties.
37.29 Waiting times are 13T/3 and 4T.
Chapter 38
38.1 £1790.85,0.487%.
38.2 (b) 16.9 years.
38.3 (b)0, (-l + Vl3)/6.
38.6 f(n) = (In «)/ln 2.
38.8 (b)wa = A3" + B(-3)».
(c) un = 3"(A cos \nn + B sin \nlt).
38.11 (a) (ii) u„ - —~n + |«2.
(b) (ii) un=\n +
(c) (iii) un - \n1.
(d) (iii) un = yj «23".
38.13 D„(l) = « + 1.
38.16 u„ = - - -(-!)".
38.17 dk = k(N-k). 38.19 s„ = i«2(l + «)2.
38.22 0< a< 1.
38.23 Oscillates between 0.4953 and 0.8124.
38.24 The periodic values of the 2-cyclc arc 0.4 and 0.8.
Chapter 39
39.1 (d)63.
39.2 The probability that the score is 7 or less is 7/12. 39.5 n(A uC) = 10.
1
39.6 (b) Ace of clubs or ace of spades drawn;
(d) any ace or any heart or any black card drawn; (f) any heart except the ace of hearts; (h) ace of hearts or any black card.
39.7 (b) 1/221; (b) 0.004 166... . 39.9 (b) 5040; (d) 7.
39.11 (a) 27 216; (c) 3360.
39.12 (b) 156 849.
39.15 270 725; 0.0.10 56... .
39.17 (a) 9/209; (c) 16/665; (d) 683/1463.
39.18 (b) 0.872; (c) 0.4.
39.19 (b) 0.37; (c) 0.82.
39.20 With the same probability of failure 0.98, probability that circuit fails is 0.963.
39.21 (b) 1/495; (c) 4/99.
39.22 Overall probability is approximately 1/53.7.
39.23 Mean number of plays to the end of the game is 2"-Vn.
Chapter 40
40.1 P(X3= 1)= 0.833.
40.2 P(Xs=6) = l/32.
40.3 Mean = 4; standard deviation = 1.633.
40.4 Mean = (a + b)!2; standard deviation =
{h-a)l(2<3).
40.6 Mean number of non-faulty components to failure is 82.33; standard deviation of the number of components to failure is 82.83.
40.7 1/29.
40.9 Probability that a bottle fails the test is 0.000 67.
40.10 (a) 0.777; (c) 0.223.
40.11 (b) 0.528.
40.13 (b)P(Z=£ 0.7) = 0.758.
40.14 On average 30% of operations take longer than 40 seconds.
40.15 Standard deviation of 1 if a = v5 and
A = 3/(20^5).
40.16 Maximum value of standard deviation is 121.6.
40.17 Probability that just two bulbs will be still working is 0.242.
Chapter 41
41.1 (b) Mean = 24.1; median = 24.5; interquartile range = 17.
41.3 Sample mean = 25.3; mode = 25.1; variance = 0.0644.
41.5 About 11 intervals.
41.6 Estimated variance of the sample is 1/12.
41.8 k, =-1.1337; k,= 1.1337.
41.9 For full data a = -0.0071; b = 18.76.
CO
m 33
CO
m r~ m O -\ m D
TJ J3 O 03
r-
m
CO
Appendices
Some algebraical rules
(a) Index laws for real numbers
(1) a° = l. (ii) apa* = ap+'i. (m) a-fi = l/ap.
(iv) [a''Y or {a*)t = a<"> (so ap/" = {ai,)l!q or (aVq)p).
(v) a»bp=(ab)p
Conventionally, a2 and represent the positive root when we are talking about real numbers (for complex numbers, see Chapter 6). For all the rules to hold in all cases, a must be positive so that ap!q is a always real number. For example, (—8)3 or V(—8) is not real: there is no real number whose square is equal to —8. But (—8)3 or^(-8)=-2.
(b) Quadratic equations
ax1 + bx + c — 0 has the solutions xt, x2 = [-h± V{b1 - 4ac)]/2a.
(i) In terms of xl and x±, the factors are
ax1 + bx + c = a{x - x,)(x - x,).
(ii) Sum and product of solutions:
„Y| + x, - —b/a,      .Yj-v, = cla.
(c) Binomial theorem
(i) If n is a positive integer (or whole number)
11    «(«-1)    ,,,    «(«-!)(« -2) ,
(a + b)n = a" + na"-lb + —--        a"-1 b1 + —---- a"-%> + --- + b"
2! 3!
a"-rbr
where the binomial coefficients are denoted by n \ n\
J) («-r)!r!
There are (n + 1) terms in this sum, and it is symmetrical in a and b. An important special case is
n[n — 1)   ,    n(n — l)(n — 2) ,
(1 + x)" = 1 —-■+• —---ix-> + ••• + x".
2! 3!
(ii) Pascal's triangle. Each entry (apart from the numerals 1) is the sum of two previous entries - that above, and that above and to the left - as illustrated by the underlined group:
n = l 11
n = 2 12 1
n=3 13 3 1
n = 4 14 6   4 1
and so on. Thus
(1 + x)4 = 1 + 4x + 6x2 + 4x3 + x\
(iii) Permutations and combinations (see Section 1.17).
n\ «'
p =_ c
(n — r)\r\
(d) Factorization
a2 — b1 = (a + b) (a — b),
a3 — b3 = (a - b) {a1 + ab + b1),
a3 + b!> = (a + b){a2-ab4-b1).
(e) Constants
e = 2.718 281 82..., 71 = 3.141 592 65..., 1 radian = 57.295 78... °, 1° = 0.01745... radians, 360° = In radians.
(f) Sums of powers of integers
ft
^ r = 1 + 2 + 3 + • ■ • + n = {n{n + 1)
r=l
n
= P + 22 + 32 + ••■ + n2 = \n(n + 1)(2« + 1)
r=l
•£r3 _ 13 +21 +33 + ... + „3 _ i„2(„ + 1)2_
Trigonometric formulae
(a) Relation between trigonometric functions
sin2A + cosM = I,
tan A — sin A/cos A;   secA=l/cosA;   cosec A = 1/sin A.
(b) Addition formulae
sin (A ± B) — sin A cos B ± cos A sin £>, cos(A ±B) = cos A cos 5 + sin A sin B, tan (A ±B) = (tan A ± tan B)/{1 + tan A tan B).
(c) Addition formulae: special cases
sin 2A = 2 sin A cos A, cos 2A = cos2A — sin2A
= 2 cos2A -1 = 1-2 sin2A, tan 2A = 2 tan A/(l - tan2A), sin 3A = 3 sin A - 4 sin3 A, cos 3A = 4 cos3A - 3 cos A.
(d) Product formulae
sin A sin B=j[cos(A-B) -cos(A + fi)], cos AcosB = |[cos(A-B)+eos(A + B)], sin A cos B = {[sin(A - B) + sm(A +B)]. sin C + sin D = 2 sin \{C + D) cos }(C - D), sin C - sin D =2 sin {{C - D) cos \(C + D), cos C + cos D = 2 cos }{C + D) cos j(C - D), cos C - cos D = -2 sin f (C + D) sin }(C - D).
(e) Product formulae: special cases
sin2A = {(l -cos2A), cos2A = j(l +cos 2A), sin3A = |(3 sin A - sin 3A), cosM = 1(3 cos A + cos 3A).
(f) Triangle formulae
(i) a + P+j=l80°.
(ii) Cosine rule: al — b1 + c1- 2bc cos (X.
..... .        ,    sin a    sin /}    sin Y
(in) i/ne rw/e: -=-=-.
a b c
(g) Trigonometric equations
In the following, n represents any integer (i.e. any whole number, positive or negative); x is in radians.
(i) sin x = 0 and tan x = 0 when x = nil; cos x — 0 when x = ^K + nK.
(ii) The following formulae show how to obtain all the solutions of certain equations when one solution has been obtained (e.g. a hand calculator or
a computer gives only one solution of sin x = —j, namely x = arcsin(—|) = -0.5236... ).
If sin OC— c, then all the solutions of sin x = c are x = nn + (—l)"a. If cos /3= c, then all the solutions of cos x = c are x = 2nK ± (3. If tan 7 = c, then all the solutions of tan x = c are x — n% + y.
(h) Hyperbolic functions
coshx = j{ex + e-*); sinhx = \(zx — e"x); tanh x = sinh x/cosh x;
sech x= 1/cosh x; coth x = cosh x/sinh x; cosech x = 1/sinh x,
sinh (x±y) = sinh x cosh y ± cosh x sinh y,
cosh(x ± y) = cosh x cosh y ± sinh x sinh y,
cosh2x — sinh2x = 1,
sinh 2x = 2 sinh x cosh x,
cosh 2x = cosh2x + sinh2x,
cosh ix = cos x; sinh ix = i sin x;
sinh"1x = ln[x+ (x2+ 1)^],
cosh-1x = ln[x+(x2-l)2l
tanh-1x = |ln[(l + x)/(l-x)]   (-1 <x<l).
Areas and volumes
(a) The area of a triangle is \bh, where b is the length of one side and h its height from that side.
(b) The circumference of a circle is 2nr, where r is its radius.
(c) The area of a circle is Tlr1, where r is its radius.
(d) The area of a circle sector is jr26, where r is its radius and 6 the angle of the sector in radians.
(e) The volume of a sphere is jKr3, where r is its radius.
(f) The surface area of a sphere is 4nr2, where r is its radius.
(g) The volume of a cone is\Ab, where h is its height and A the cross-sectional area of its base.
(h) The area of an ellipse is Tiab, where a and b are the lengths of its semi-axes.
(i) The area of a regular w-sided polygon of side-length a is \nal cot(7t/«).
CO LU
Ü
LU
a
D. <
A table of derivatives
c (constant)
x" (n any constant)
e"
k' (k > 0) In x (x > 0) sin ax cos ax tan ax cot ax sec ax cosec ax arcsin ax arccos ax arctan ax sinh ax-cosh ax tanh ax sinh~1ax cosh 'ax
tanh
ax
u(x)v{x)
u(x) v[x)
1
v (x) >'(«(x))
y(y(«(x)))
dy dx
0
«x"~2 a e"A kx In
X"1
a cos ax
—a sin ax
a/cos2ax
—a/sin2x
(a sin ax) I cos1 ax
—(a cos ax)/sin2ax
a/(l -a2x2p
-a/(l - a2x2p
a/(1 +a2x2)
a cosh ax
a sinh ax
a/cosh2ax
a/(l + a2x2)'
a/(a2x2 - 1)3
4/(1 - a2x2)
c\v du U— + V— dx dx
JLj
f2V dx
f2 dx
dy du du dx
dy dv du dv du dx
dx
Tables of indefinite and definite integrals
fix)
xm (m*-l)
kx (k > 0)
In x (x > 0)
sin ax
cos ax
tan ax
cot ax
sec ax
cosec ax
arcsin ax arccos ax arctan ax
sinh ax cosh ax tanh ax
ll(x2 + a2)
l/(x2 - a1)
Ilia1 -x2)^ \/ia2 + x1)i llix1 - a2)^ x e" x cos ax x sin ax x In x e'x cos bx e"v cos /;x
/(x) dx (C is an arbitrary constant.)
1
+1+C m + 1
In | x | + C", or In | Cat | (IIa) &X+C kxl\nk + C x In x — x + C —(l/«) cos ax + C (1 /a) sin ax+ C
—(l/«) ln|cos ax| + C or— (1/a) ln|C cos ax\
(11 a) In | sin ax \ + C or (1/a) ln| C sin ax|
-(1/2«) In[(1-sin ax)l(\ + sin ax)] + C
(1 /2a) ln[(l - cos ax) 1(1 + cos ax)] + C
(1/a) (1 — a2x2)~2 + x arcsin ax + C —(lla) (1 — a2x2)2 + x arccos ax + C —(l/s)ln(l - a2x2)2 + x arctan ax + C
(1/a) cosh ax + C (lla) sinh ax + C (1/a) In {cosh ax} + C
(lla) arctan(x/a) + C
(1/2«) In | (x - a)l(x + a) | + C or (lla) Vin[rl(x/a) + C
arcsin(x/a) + C (or — arccos(x/a) + C)
(1/a) sinrr'(x/a) + C or ln[x + (x2 + a2)i] + C
\n[x + (x2-a2)i] + C
(l/a2)(«x- 1) eax+C
(1/a1) (cos ax + ax sin ax) + C
(1/a2) (sin ax — ax cos ax) + C
4-x2 In x - -j-x2 + C
[1 /(a1 + b1)] efx(a cos bx + b sin bx) + C [l/(a2 + b2)] c™(—b cos bx + a sin fox) + C
> -a -o m
O m
CO
A table of definite integrals
dx 71
j 0 a2 + x1 2a
: dx -
cos x dx = 1
o
(0,	m ^ »1
IK	m = n\
-(?'	
Is".	m — n
	
\2ml(	m1 — «2
[m, n positive integers) cos mx cos nxdx = \ ^ " \   [m, n positive integers)
)
.10, m + n even I
sin mx cos nx dx = 1      .   2     , s      , , f   (w, »positive integers)
xnc'dx = n\   (« = 0,1,2, ... )
o
e~"I_dx-- I—   (a > 0) o 2-V a
e ax cos fox dx
~TT77 (*>0) <r + ft2
&
e "* sin fox dx = —;——   [a > Ü) a + tr
Gradshteyn and Ryzhik (1994) is a useful source of hundreds of indefinite and definite integrals.
Laplace transforms, inverses, and rules
In the following tables, n and in represent a positive integer or zero. The constants k and c arc arbitrary unless otherwise indicated.
	Transforms			Inverses
fit)	F(s) =	e-s>f(t)dt 0	F(s)	fit)
V	n\		1	
				(m-1)!
efa	1		1	
	s-k		s-k	
t" ekt	n\		1	
	is - k)	111	(s - kf	(m-\)\
cos kt	s		s	cos
	s1 + k1		s1 +k2	
sin kt	k		1	1 . , — sin kt k
	s1 + k1		s2 + k2	
t cos kt	s2-k2		s1 - k1	t cos kt
	(s1 + k1)1		(s2 + k2)2	
t sin kt	Iks		s	1 ,
	(s1 + k1)1		(s2 + k2)2	— t sin &f
H(t-c) [c > 0)	e~"/s		e'"/s (c > 0)	H(f-c)
8{t-c) (c>0)	e-cs		e"" (c > 0)	8(t-c)
Summary of rules: In the following rules, F(s) <-> /"(£).
■S'ca/e rw/e (24.5)
.S'^'/if rule, or multiplication by eh (24.7)
Powers oft (24.8)
Derivatives (24.12) Delay rule (24.15)
1 /s as a« integration operator (25.1)
l (t\
and   F(ks)^-f - (k>0).
If   is any constant, ek'f(t) O F(s — fe).
If w is a positive integer, then t"f{t) <-» (—1)"
d"F(s) ds"
ffi ^ sF(5) _ A0))   fPf|) ^ j2f(s) _ s/.(0) _ f{0) at at2
If c > 0, then e""F(s) <-» f(t - c)H{t - c) (where H is the Heaviside unit function).
f
1
If F(s)<-> fit), then -F(s)<-»
/■(t) dr.
Convolution theorem (25.11)        If g(t) <-> G(s) and /"(?) <-> F(s), then
F(s)G(5) <H>
g(f - T)f{T) dz
g(T)f(t - T) dT
to
LU
O
O Z LU
CL Q-<
ponential Fourier transforms and rules
Signal (time function)
Transform (frequency distribution)
Fourier transform pair	x(t)=     X(f) e'W'df	X(/') = j  x(t) c'1^'dt
Linearity	Axx{t) + Bx2(t)	AXt(f) + BX^f)
Time scaling	x(At)	\A\-íX(A-'f)
Time reversal	x(-l)	X(-f)
Time delay	x(t-B)	X(f) e.-ilKHf
Frequency scaling		X(Cf)
Frequency shift	x{t) e'2nD'	X(f-D)
Modualtion	x(t) cos 2nKt	j[X(f+K)+X(f-K)\
	x(t) sin 2nKt	j\i[X(f+K)-X(f-K)]
Differentiation	dx(t)/dt	{Hnf)X(f)
	d"x{t)/dť	mf)"X(f)
Duality	X(t)	x(-f)
Convolution	xl(u)x1(t - u) du = x,(í) * x2(t)	Xi(f)XÁf)
Multiplication
x,(i — u)x,(u) du xt(t)x2{t)
Periodic function xr{t)      xP{t) (period T)
X,(f - v)X2{v) dv X2(v)X2{f -v) dv £x„8(/- nfu), where/i =1/T,
Short table of Fourier transforms
Signal		Transform	Signal	Transform
U(t) = H(t - 4)	- H(ř + \)	sine f	1	71 e-i,:'!
sine t			1 + i2	
[1 + f,	-1< t < 0			
A(f)= 1-ř,	0<r<l	sine2/	5(f)	1
K	elsewhere		1	
sinc2i		Mf)	cos 2nfut	{i8(/+/;,) + 5(/'-f0)j
e 'H(i)		l/( 1 + 1271/")	sin 27l/nř	
ře"řH(ř)		1/(1 + Í27T/')2	LUT(f)= £s(r-«T)(T>0)	f0mh(f) (f0 = i/T)
e-l'l		2/(1 + 4tcY2)		
Probability distributions and tables
(a) Distributions, means, and variances
(i) Discrete distributions
	Distributi on	Probability	Mean (fi)	Variance (<72)
	Binomial	n\p'q"~' (n — r)\r\	np	npO -p)
	Geometric	(l-p)'-'p	1	\-p P1
	Poisson	A" e-1	A	A
	Pascal		fc P	k(l - p) P1
	Hypcrgeometric	<■(':. ,	»/?	nwb(b + w + n)
				[w + bf(iv + /;-!)
	(ii) Continuous distributions			
	Distribution Density		Mean (/I)	Variance (CT2)
	Exponential	0,        x < 0	1 A	1
	Uniform	rl/{b-a), a<x<b 0, elsewhere	{(« + bf	
	Standardized		0	1
	normal [Z			
>
TJ
"D
m Z O
o m co
(b) Cumulative normal distribution tables
Standardized cumulative normal distribution giving the values of
0(X) =
1
V2jt
e-i'2di
for 0 =S x s£ 3.0 at 0.01 intervals. For x < 0, 0{x) can be calculated from &(—x)
1-0(X).
x      0 1234 5 6789
0	0	0.5000	0.5040	0	5080	0	5120	0	5160	0	5199	0	5239	0	5279	0.5319	0	5359
0	1	0.5398	0.5438	0	5478	0	5517	0	5557	0	5596	0	.5636	0	5675	0.5714	0	5753
0	2	0.5793	0.5832	0	5871	0	5910	0	5948	0	5987	0	6026	0	6064	0.6103	0	6141
0	3	0.6179	0.6217	0	6255	0	6293	0	6331	0	6368	0	6406	0	6443	0.6480	0	6517
0	4	0.6554	0.6591	0	6628	0	6664	0	6700	0	6736	0	6772	0	6808	0.6844	0	6879
0	5	0.6915	0.6950	0	6985	0	7019	0	7054	0	7088	0	7123	0	7157	0.7190	0	7224
0	6	0.7257	0.7291	0	7324	0	7357	0	7389	0	7422	0	7454	0	7486	0.7517	0	7549
0	7	0.7580	0.7611	0	7642	0	7673	0	7704	0	7734	0	7764	0	7794	0.7823	0	7852
0	8	0.7881	0.7910	0	7939	0	7967	0	7995	0	8023	0	80.51	0	8078	0.8106	0	8133
0	9	0.8159	0.8186	0	8212	0	8238	0	8264	0	8289	0	8315	0	8340	0.8365	0	8389
1	0	0.8413	0.8438	0	8461	0	8485	0	8508	0	8531	0	8554	0	8577	0.8599	0	8621
1	1	0.8643	0.8665	0	8686	0	8708	0	8729	0	8749	0	8770	0	8790	0.8810	0	8830
1	2	0.8849	0.8869	0	8888	0	8907	0	8925	0	8944	0	8962	0	8980	0.8997	0	9015
1	3	0.9032	0.9049	0	9066	0	9082	0	9099	0	9115	0	9131	0	9147	0.9162	0	9177
1	4	0.9192	0.9207	0	9222	0	9236	0	9251	0	9265	0	9279	0	9292	0.9306	0	9319
1	5	0.9332	0.9345	0	9357	0	9370	0	9382	0	9394	0	9406	0	9418	0.9429	0	9441
1	6	0.9452	0.9463	0	9474	0	9484	0	9495	0	9505	0	9515	0	9525	0.9535	0	9.545
1	7	0.9554	0.9564	0	9573	0	9582	0	9591	0	9599	0	9608	0	9616	0.9625	0	0633
1	8	0.9641	0.9649	0	9656	0	9664	0	9671	0	9678	0	9686	0	9693	0.9699	0	9706
1	9	0.9137	0.9719	0	9726	0	9732	0	9738	0	9744	0	9750	0	9756	0.9761	0	9767
2	0	0.9772	0.9778	0	9783	0	9788	0	9793	0	9798	0	9803	0	9808	0.9812	0	9817
2	1	0.9821	0.9826	0	9830	0	9834	0	9838	0	9842	0	9846	0	9850	0.9854	0	9857
2	2	0.9861	0.9864	0	9868	0	9871	0	9875	0	9878	0	9881	0	9884	0.9887	0	9890
2	3	0.9893	0.9896	0	9898	0	9901	0	9904	0	9906	0	9909	0	9911	0.9913	0	9916
2	4	0.9918	0.9920	0	9922	0	9925	0	9927	0	9929	0	9931	0	9932	0.9934	0	9936
2	5	0.9938	0.9940	0	9941	0	9943	0	9945	0	9946	0	9948	0	9949	0.9951	0	9952
2	6	0.9953	0.9955	0	9956	0	9957	0	9959	0	9960	0	9961	0	9962	0.9963	0	9964
2	7	0.9965	0.9966	0	9967	0	9968	0	9969	0	9970	0	9971	0	9972	0.997.3	0	9974
2	8	0.9974	0.9975	0	9976	0	9977	0	9977	0	9978	0	9979	0	9979	0.9980	0	9981
2	9	0.9981	0.9982	0	9982	0	9983	0	9984	0	9984	0	9985	0	9985	0.9986	0	9986
3	0	0.9987	0.9987	0	9987	0	9988	0	9988	0	9989	0	9989	0	9989	0.9990	0	9990
Table giving x for specified values of <P(x) for 0.50 '  <P(x) '  0.99 at 0.01 intervals
0(X)	x		X	*(*)	X
0.50	0.0000	0.67	0.4399	0.84	0.9945
0.51	0.0251	0.68	0.4677	0.85	1.0364
0.52	0.0502	0.69	0.4959	0.86	1.0803
0.53	0.0753	0.70	0.5244	0.87	1.1264
0.54	0.1004	0.71	0.5534	0.88	1.1750
0.55	0.1257	0.72	0.5828	0.89	1.2265
0.56	0.1510	0.73	0.6138	0.90	1.2816
0.57	0.1764	0.74	0.6433	0.91	1.3408
0.58	0.2019	0.75	0.674.5	0.92	1.4051
0.59	0.2275	0.76	0.7063	0.93	1.4758
0.60	0.2533	0.77	0.7388	0.94	1.5548
0.61	0.2793	0.78	0.7722	0.95	1.6449
0.62	0.3055	0.79	0.8064	0.96	1.7507
0.63	0.3319	0.80	0.8416	0.97	1.8808
0.64	0.3585	0.81	0.8779	0.98	2.0537
0.65	0.3853	0.82	0.9154	0.99	2.3263
0.66	0.4125	0.83	0.9542		
Dimensions and units
Physical quantities of different types, such as acceleration, force, momentum, electrical potential, can be classified by expressing them as simple combinations of certain primary dimensions such as mass, length and time. These expressions determine how we can state the magnitude of a physical quantity — for example any velocity can be expressed in metres per second, but never in metres per kilogram. Five primary dimensions provide a basis sufficient for all common purposes. Their names, the algebraic symbols denoting their dimension, and appropriate units (the international (SI) system) are shown in the following table.
Basic quantity	Dimension symbol	SI unit	Unit symbol
length	L	metre	m
mass	M	kilogram	kg
time	T	second	s
electric current	I	ampere	A
absolute temperature	e	Kelvin	K
We can now assign dimensions to any derived physical quantity, which classifies it without indicating its magnitude. For example, the velocity at any moment of any particle, substance, electromagnetic wave, etc., could in principle be measured as {a distance travelled)/[time taken). Symbolically we write:
[velocity] = IT-1,
where the square brackets mean 'the dimension of. The form LT_1 of the right-hand side indicates that an appropriate SI unit of measurement would be metres per second. The following table comprises mechanical and electromagnetic quantities, their dimensions, and conventional SI terms for certain special units of measurement. Notice how known dimensional forms may be multiplied and divided to obtain more complicated ones.
Derived quantity	Dimension	SI unit	Symbol	Name
angle	dimensionless	_	rad	radian
velocity (displacement/unit time)	LT"1	m s_I	-	-
acceleration (velocity/unit time)	LT"2	m s-2		-
force (mass X acceleration)	MIT"2	kg m s"2	N	newton
momentum (mass x velocity)	MLT 1	kg m s'1	-	-
moment of momentum	MLT1	kg m2 s"1	-	-
pressure (force/unit area)	ML-'T-2	N m 2	Pa	pascal
work, energy (force x distance)	ML2T 2	Nm	J	joule
area	L2	m2	-	-
volume	L3	m3	-	-
power (work/unit time)	MLT3	Js '	W	watt
angular frequency (radian/unit time)	T-i	S"1	Hz	hertz
charge (current X time)	IT	As	C	coulomb
potential (work/unit charge)	ML/TT1	J c-'	V	volt
resistance (potential/unit current)	ML2TT2	V A"1	Q	ohm
magnetic flux (work/unit current)	ML2T T1	JA"3	Wb	weber
inductance (magnetic flux/unit current)	ML2T-T2	Wb A"'	H	henry
>
TJ TJ
m Z
o
o m w
For comprehensive tables of units and constants, consult Kaye and Laby (1995).
If two physically meaningful expressions are equal, then both sides must obviously have the same physical dimensions. This often provides a useful check on a calculation. Also, in any expression containing the sum of two or more terms, the terms must all have the same dimensions if it is to make any physical sense. For example, expressions equivalent to the form (energy + momentum), or [current + voltage) can have no physical significance. However, in such cases the dimensions of any letters used as constant factors must not be overlooked: the expression [momentum + (k X energy)) could be meaningful provided [k] = TL"1.
The dimensions of quantities that appear as derivatives and integrals are treated in the following way. Suppose for example that /. is time ([t] = T) and x(t) is a function representing displacement (\x\ = L). Then
dx
LT"
dt_
and so on. Also x(t) dt
dzx
LT-2,
: LT, and
t dt
= T2.
These follow from the definition of the integral as a sum.
Dimensional analysis is helpful in checking the validity of equations. For example, in the pendulum equation (see eqn 20.22)
—--h^sin 69=0 dt1 I
all terms should have the same dimensions, which is true since
d29
dt2
= T"2 and
sin 6
LT~2L~
where g is the acceleration due to gravity, / is the length of the pendulum, and the angle 6 and sin f}are dimensionless. Physically the equation
d20
with the same definition of symbols could not represent a general physical law because the dimensions of the two terms are different.
Dimensionless analysis indicates how equations can be simplified by making them dimensionless. In the pendulum equation above, let T= W(g//). Then the dimensionless pendulum equation becomes
d2e
dr2
+ sin 0=0
which includes pendulums of all lengths, in any uniform gravitational field.
Further reading
iimiiiumhiiihhi 11 iinmiiwwlill»iii»i»i»iiin—mi(««ni—nii>wiiiÉiiimiiii«ii <     nium—iii»iuiiimi ■ in iimiii^ii——iiwi—ii——P—iiiiii im i »min i ...... i i n     u.......n 111 in Nil Ii ill i n i n 11   íl      i 11 in i i íl i........i ........inii ......mi mm iiiiiimwiiiniw i
Abell, M.L. and Braselton, J.P. (1992) Mathematica by Example, Academic Press, San Diego.
Blachman, N. (1992) Mathematica: A Practical Approach, Academic Press, San Diego. P)oyce, W.E. and DiPrima, R.C. (1997) Elementary Differential Equations and Boundary
Value Problems (6th edn), Wiley, New York. Gamier, R. and Taylor, J. (1991) Discrete Mathematics for New Technology, Adam
1 lilger, Bristol.
Gradshteyn, LS. and Ryzhik, I.M. (1994) Table of Integrals, Series, and Products (5th
edn). Academic Press, San Diego. Grimmett, G.R. and Stirzakcr, D.R. (2001), Probability and Random Processes (3rd edn),
Oxford University Press. Jordan, D.W. and Smith, P. (2007a) Nonlinear Ordinary Differential Equations (4th
edn), Oxford University Press. Jordan, D.W. and Smith, P. (2007b) Nonlinear Ordinary Differential Equations:
Problems and Solutions, Oxford University Press. Kaye and Laby (1995) Tables of Physical and Chemical Constants, National Physical
Laboratory (16th edn) (available online at www.kayelaby.npl.co.uk). Montgomery, D.C. and Rungcr, G.C. (1994) Applied Statistics and Probability for
Engineers, Wiley, New York. Ráde, L. and Westergren, B. (1995) Mathematics Handbook for Physics and Engineering,
Studentlitteratur, Lund. Riley, K.F., Hobson, M.P. and Bcncc, S.J. (1997) Mathematical Methods for Physics and
Engineering, Cambridge University Press. Roberts, G.Ľ. and Kaufman, H. (1966) Table of Laplace Transforms, Saunders,
Philadelphia.
Seggern, D.H. von (1990) CRC Handbook of Mathematical Curves and Surfaces, CRC Press, Baton Roca.
Skcel,  R.D.   and   Keeper,  J.B.   (1993)   Elementary  Numerical  Computing with
Mathematica, McGraw-Hill, New York. Whitelaw, T. A. (1983) An Introduction to Linear Algebra, Blackic, Glasgow. Wilson, R.J. and Watkins, J.J. (1990) Graphs: An Introductory Approach, Wiley, New
York.
Wolfram, S. (1996) The Mathematica Book (3rd edn), Wolfram Media/Cambridge University Press.
Zwillinger, D. (1992) Handbook of Differentials Equations (2nd edn), Academic Press, Boston.
Index
Pages of the main topics are given in heavy type for quick reference.
Abscissa 6 absolute value 6 acceleration 68, 212
polar components 216
radial 216
transverse 216
vector 213 adjacency matrix 837 adjoint (adjugate) matrix 189, 190
admittance 536 algebra, Boolean 801-813 (see also Boolean algebra) algorithm, (numerical) 118,464
(see also approximation) amplitude 22,414
complex 453 angle 16
degree 16
dimension 959
polar 12
radian 16,949 angular (circular) frequency 22,
415,428 angular momentum 258 angular spectrum function 612 angular velocity 258 antiderivative 307-318
and area 314
bracket notation 316
composite 317
table of 313 antidifferentiation 307-318 (see
also antiderivative) approximations
algorithm 118, 464
bisection method 122
Euler method 463, 499
Gauss—Seidel method 273
incremental 115,645, 683
iterative process 118
Jacobi method 274
lineal element diagram 461, 926
linear 646
Newton's method (for
equations) 116-119 rectangle rule 322, 347 Simpson's rule 355, 925 step-by-step 108 Taylor polynomials 125, 922 trapezium rule 347,924 arccos, arcsin, arctan functions 25
arc length 355
area (see also integrals)
analogy for integrals 333, 346
as a definite integral 323
geometrical 344
as a line integral 761
parallelogram 248
in polar coordinates 345
signed 314,320, 327
as a sum 285
of a surface 343
table 951
trapezium rule 347
of a triangle 951 Argand diagram 144
imaginary axis 144
parallelogram rule 145
for phasors 443
real axis 144 argument
complex number 146
function 13
principal value 146 asymptote 11, 109,113,114 attenuation 611 attractor 858
strange 858 augmented matrix 263 autonomous differential
equations 481 axes, cartesian 6
abscissa 6
coordinates 6
left-handed 6,198, 246
oblique 257
ordinate 6 origin 6
right-handed 6,198,246 rotation of 223, 226-229
bar chart 888 basis
differential equations 385-390
vectors 210 Bayes' theorem 880 beam problem 354 beats 431-437
frequency 432
period 432 Bernoulli
equation 478
trial 887 binary
operation 801
set 789,802 binomial distribution 887-888
mean 889
Poisson approximation 893
variance 890 binomial theorem 51—54, 120, 948
coefficient 51
Pascal's triangle 52
Taylor series 131 bins (statistics) 904 bipartite graph 832
complete 832 bisection method 122 block diagram 827
reduction 828 Boolean algebra 801-813
absorption laws 802
algebra 802
AND gate 804
associative laws 802
binary addition 813
binary operation 801
binary set 802
Boolean expression 803
commutative laws 802
complement 801
complement laws 802
conjunction 804
de Morgan's laws 802
disjunction 804
disjunctive normal form 808
distributive laws 802
duality principle 812
exclusive-OR-gate 808
expression 803
EXOR gate 808
identity laws 802
join 801
logic gates 803
logic networks 805
logically equivalent gates
812 meet 801 NAND gate 805 negation 804 NOR gate 805 NOT gate 804 OR gate 804 product 801 reflexive law 802 sum 801
switches in parallel 810
switches in series 810
switching circuits 809
switching function 810
truth table 803
truth table, inverse 808
variables 802 box plot 906, 930
interquartile range 907
median 907
quartiles 906
outliers 907
whiskers 907 branch (graph theory) 821
Capacitor 447, 528
complex impedance 447, 533
phasor 446 cardinality (of a set) 798 cardioid 57, 355, 920 carrier wave 432, 585, 598 caustic 707
Cayley-Hamilton theorem 302
cdf (cumulative distribution
function) 895 cells (statistics) 904 central limit theorem 911 centre (phase plane) 484, 485,
492,497 centre of mass 348, 350 centroid 349,363 chain rules 86, 91, 101, 631, 664,
668,676 more than one parameter 668,
676
one parameter 664, 668 chaos 857,861,865,929 characteristic equation
difference equations 850
differential equation 385-391
matrices 279 circle 10
area 951
cartesian equation 10 circumference 951 vector equation 207 circuits(electncal) 105, 380,418,
446-453, 823-827, 838,
839,878 balanced bridge 451 cutset method 823 LCR418
Laplace transform nethods 535
parallel 448, 878
RL380
scries 448, 878
signal flow graphs 827
switching 809 cobweb diagram 847-849 cofactor 180
combinations 49-51, 949
common ratio (geometric series) 43
compatibility
linear equations 267 complement (of a set) 781 complementary function (see also difference equations; differential equations) difference equations 852 differential equations 405 complete graph 817 completing the square 11, 140, 367
complex impedance 446-451 (see also impedance) complex numbers 140-156 (see also Argand diagram) argument 146 conjugate 142,143 de Moivre's theorem 150 difference 142 division 142
exponential form 148, 151 Euler's formula 149 imaginary part 141 logarithm 142 modulus 141, 145 ordered pair 144 parallelogram rule 145 polar coordinates 146 principal value 146 product 142 quotient 142 real part 141 reciprocal 142 rules for 141 standard form 141 sum 141
compound interest 122, 842-843 conditional probability 875-877 cone 241, 625
surface area 342
volume 342, 951 conic sections 12 conjunction 804 conjugate, complex 142,143 connected graph 817, 820 conservative field 752-759,775
potential 754, 775 continuity equation 772 contour map 625-626, 636, 927 convergence
of infinite series 129
of integrals 330 convolution 541, 927 {see also Fourier transform; Laplace transform; z-transform)
discrete 552
Fourier transform 535-538, 956
Laplace transform 541, 726, 927
memory and 544 theorem 541,535,726 z-transform 490
coordinates, three-dimensional (see also axes)
cartesian 623
curvilinear 672
cylindrical polar 777
orthogonal systems of 675
paraboloidal 785
rotation of 226-229
spherical polar 780 coordinates, two-dimensional (see also axes)
cartesian 6
origin 6
orthogonal systems 675
polar 28-30
rotation 223 coplanar vectors 218, 251 cosh function 37
Taylor series 131 cosine function 18 (see also
trigonometric functions)
antiderivative 313
derivative 76
exponential form 150
Taylor scries 130 cosine rule 58,116, 950 cosine/sine transforms 587—590
inverse 589
at a jump 589 counting index(series) 43 Cramer's rule 260,262,270 cross product (see vector
product) cumulative distribution
function(cdf) 895 curl 773-776
in curvilinear coordinates 780
determinant formula 774
identities 785 curvature 238, 243
centre of 122
radius of 123,239 curves
angle between intersecting 658 asymptotes 11,109,113,114 caustic 707 chord 62-65 convex/concave 239 curvature of 238,243 envelope 475,702, 928 gradient 62 length 355
normal to 238,657, 658
orthogonal systems of 928 parametric equations 95, 664 point of inflection 93, 238 radius of curvature of 123,239 sketching 108-114 slope 62-65 tangent line 62-66 tangent vector 212 curvilinear coordinates 672 curl 780
cylindrical polars 777
divergence 780
elliptic system 674
gradient 780
paraboloidal 785
scale factor 780
spherical polars 780 cutset (graph theory) 822, 929
fundamental 824 cycle(graph theory) 820 cylindrical polar coordinates 704,777
damper 418
damping 419 critical 439 heavy 420 weak 420
dash notation for derivative 100
dcadbeat 420, 488
decay, radioactive 36,393
definite integral 320-338
degreefof angle) 16
degree (of a vertex) 817, 818
delay rule (second shift rule) 522
del (grad) operator 659
delta function (impulse function) 530,599 and discrete systems 546 Fourier transform of 599 and Heaviside unit function 532
Laplace transform of 531 dc Moivrc's theorem 150 de Morgan's laws 795, 802 derivative, directional (see
directional derivative) derivative, ordinary 65 (see also derivative, partial) and antiderivative 307 chain rule 86, 91
dash notation 100
definition 65
dot notation 215, 480
function of a function rule 86
higher order 77, 102
implicit 93
and incremental
approximation 115 index notation 125 of inverse functions 94 logarithmic 92 material 690 notations 65, 100 parameter, in terms of 95 of polynomials 126 of product 83, 101 of quotient 85, 101 and rate of change 67 of reciprocal 85, 101 second 79 of sums 70
table of derivatives 76, 91, 952
total 665
of vectors 213
of f(ax + b) 90
of ex 75
of In x 76
of cos x, sin x 75
ofx"69, 89 derivative, partial 627
higher 629
mixed 630
second 629 determinants 173,175,179-190, 922
2x2 173,179
3x3 175,180
cofactor 180
cofactor, sign rule 181
expansion by first row 180
expansion, general 185
factorization 191, 922
Jacobian 728
minor 303
notation 179
product 188
rules 182-188
suffix permutation 180
tridiagonal 192
zero 186 diagonal dominance 274 diagonalization of a matrix 286-289, 923
difference (sets) 794 difference equations 842-861
attractor 858
bifurcation 857
chaos 857
characteristic equation 850 cobweb 847-849 complementary function 852
compound interest 843 constant coefficient 849 difference 843 equilibrium 846 Feigenbaum sequence 858 first-order 847 fixed point 846 forcing term, table generating function homogeneous 849-852 inhomogeneous 852-853 linear, constant coefficients logistic equation 845,
854-858,861 order 845
particular solution 852-853 period-2 cycle 858 period-3 cycle 858, 861 pcriod-4 cycle 858 period doubling 856 recurrence relation 843 stability 847, 854 strange attractor 858 z-transform 556 differential-delay equation 560 differential equations, first order 379-382, 407-410, 460-479 Bernoulli equation 478 change of variable 473 and differentials 469-473 direction field 461 direction indicators 46 I energy transformation 473 Euler numerical method 463 graphical method 460 integrating factor 408-410 isoclines 462
lineal-element diagram 461 logistic 478
numerical solution 473
separable 466-469,474 singular solutions 468,475 solution curves 461
variable coefficients, linear 407
differential equations, linear constant coefficient 379-412
basis 385,388 characterstic equation
385-391 complementary function 405 damped oscillator 390 first-order 382 forced equations 395-407 general solution 382, 386, 404,
405, 420 harmonic forcing 399 homogeneous (unforced)
equations 379 initial conditions 384, 391 particular solutions 395-404 second-order, unforced
384-392 second-order, forced 395-412 superposirion principle 399 unforced equations 379-394 differential equations, nonlinear
(qualitative methods)
480-502 autonomous 481 centre 484,494 direction of paths 488,
492-493 Duffing equation 502 equilibrium point 484,493 Euler's method 500 initial value problem 481 instability 486 limit cycle 497 linearization 494 linearized systems,
classification of 494, 496 node 488,494 numerical method 499 orbit (phase path) 484 periodic motion 484 phase diagram 482 phase paths (trajectories,
orbits) 484, 488, 493 phase plane 483 saddle 484,494 self-similar systems 495 separatrix 491 spiral 487, 494 stability 486,487
state of the system 482
trajectory (phase path) 484
van der Pol equation 480, 492 differential form 469, 679
for differential equation 469-473
integrating factor 472
and line integals 744
perfect 472, 744
table of 470 differentiation 61-80 (see also derivative)
chain rule 86,91
function of a function rule 86
implicit 93
of integral with respect to
parameter 640 of inverse functions 94 logarithmic 92 partial (sec also derivative,
partial) product rule 83, 101 quotient rule 85, 101 reciprocal rule 85 reversing 307 of vecrors 213 diffraction 417,608-618 angular spectrum 610 array distribution 617 attenuation 611 convolution 6 1.6 interference 615 pattern 610
phase change on ray 609,611
radiating strip 608
radiation 613
radiation rules 614
source distribution 612 digraph (directed graph) 816
weighted 828 dimensions 959-960 directed graph 816 directed line segment 198 direction cosines 225 direction ratios 229, 230 directional derivative 651-654,
661,692-696 discrete systems 545-558 (see also z-transform)
impulsive input 545
input/ouput 545
sampling 546
signal 545
time invariant 545
transfer function 549 disjunction 804 disjunctive normal form 808 dispersion 436 displacement 193
relative 195 displacement vector 195
addition 197
components 196 distance 7
of point from plane 234 distribution, sampling 908 distributions, probability (see
probably distributions) divergence (of a vector field) 764, 779,780
in curvilinear coordinates 780
in cylindrical polars 779
identities 785
in spherical polars 781
theorem 771 divergent series 129 dodecahedron 833 Doppler effect 437 dot product (see scalar product) double integration 708-734
change of variable 727-731
changing order, constant limits 712
changing order, non-constant limits 715
constant limits 709-713
double integrals 717
inner integral 709
Jacobian 727
Jacobian, inverse 734
non-rectangular regions 713
outer integral 709
polar coordinates 721
repeated integral 709
region of integration 718
separable type 724
signed volume analogy duality principle 596, 800, 812 Duffing equation 502 dummy variable 13
e, numerical value 949 echelon form 264 edge (of graph) 814
eigenvalues 279-304
characteristic equation 279 complex 280
in differential equations 496
orthogonal 293
repeated 283
vibrating system 298
zero 283 eigenve ctors 279-304
in differential equations 279
orthogonal 293
for repeated eigenvalues 283 elementary row operations 262 ellipse 11
area 951
parametric equations 99 polar coordinates 29 semi-axes 11 empty set 791
energy transformation 475, 501 envelope 475, 702 928, equilibrium (forces) 236 equilibrium point 484, 493
centre 484, 494
node 488,494
saddle 484,494
spiral 487, 494
stability of 487 errors 649,683—685 (see also
approximation) escape velocity 479 estimate (statistical) 906, 909 estimator of parameter 909
biased/unbiased 909
sample mean 905, 911
sample variance 911
standard error 910 ethanol molecule 816 eulerian graph 821 Euler's constant 926 Euler's formula (complex
numbers) 149 Euler's method (differential equations) 463, 499 Euler's theorem (graph theorem)
832 events 866
exhaustive 869
independent 877
intersection 869
mutually exclusive 869
partitioned 870
union 868
exp(x) (see exponential function) expected value (mean,
expectation) 889, 897 exponential distribution 897, 957 exponential function 30—33
derivative 75
doubling period 35
doubling principle 35, 36
growth, decay 32, 35
half-life period 36
Laplace transform of 507, 509
limit of ax"e~" 110
Taylor series 130
value of e 32 expression 5
face (of a graph) 832 factorial function 48, 78, 372 feedback 827 Feigenbaum sequence 858 fibonacci sequence 860 field (see also vector field)
conservative 752-759,775
intensity 753
potential 754,775 fixed point 846
stability 847-849 fluid flow 662, 690,706, 772,774
material derivative 690 flux 770
focal length 121, 662 force
at a point 235
components 236
equilibrium 235
moment 251, 254
resultant 235 Fourier coefficients 564—576 Fourier series 562-585
average value 567
carrier wave 585
coefficients 566
complex coefficients 580
cosine series 572
even functions 572
extensions 576
fundamental frequency 562
Gibbs' phenomenon 927
half-range series 574-576
harmonics 562
at a jump 572
Laplace transform of 585 odd functions 572 Parseval's identity 585, 608 period 271 568 period T 564
periodic function 563, 567 pitch 562 sawtooth wave 5 sine series 572 spectrum 577 switching functions 573 symmetry 572 two-sided 579-582 Fourier transforms 587-620, 956 (see also diffraction) convolution 601-605 cosine transform 586, 589 definitions 588, 589, 591 delta function 599 of derivative 596 Dirac comb 605 duality 596 exponential 591, 592 of exponential function 595 Fourier transform pair 591 frequency distribution
function 588 frequency scaling 596 frequency shift 596 fundamental frequency 587 generalized functions 600 inverse transform 589, 591 jump discontinuity 591 modulation 596 notations 527 Parscval theorem 608 periodic function 599 Rayleigh's theorem 607 rules, table of 596, 956 shah function 605 sidebands 598 signal energy 607 sine function 593, 594 sine transform 587, 588 spectral density 588 table 956 time sealing 596 top-hat function 593, 594
triangle function 605 frameworks 834-835 bipartite graph 834 minimum bracing 834
frequency 22, 414 angular 22, 415 domain 451 forcing 376
polygon (statistics) 903 friction 418
function 12 (see functions of one,
two and N variables) complementary 405, 852 generating 860 implicit 12 functions of one variable 12-35
(see also derivative;
differentiation) argument 12 delta 530, 599 dependent/independent
variables 12 discontinuous 14 even 13
exponential 30, 33 harmonic 21, 413 Heaviside 14 hyperbolic 36, 153, 951 implicit 12 impulse 530
incremental approximation
115,645,683 input/output 12 inverse 23-25 inverse hyperbolic 38 inverse trigonometric 25-28 logarithm 33 maximum/minimum 102 mean value 339 odd 13 periodic 22 point of inflection 93 rational 14 signum (sgn) 15 stationary points 102 switching 810 translation of 13 trigonometric 17-22, 25-27,
949 (table) unit step 14 functions of two variables
623-642,645-683 chain rule, one parameter
664-665 chain rule, two parameters
676-679 contour map 625
curves, angle between curvilinear coordinates 672 dependent/independent
variables 623 depiction of 624 derivatives, mixed 630 directional derivative 652, 681 errors 648-650 gradient vector 659 higher derivatives 629 implicit differentiation
654-656, 666 incremental approximation 645 Lagrange multiplier 667-672,
681
least squares method 638-640
level curves 625
linear approximation 646
maximum/minimum 635
maximum/minimum,
restricted 667-672 normal to a curve 658, 660 normal ro surface 632 orthogonal systems of curves
656
partial derivatives 627 saddle point 637 stationary points, Lagrange
multipliers 670 stationary points, restricted
667
stationary points, tests for 637 steepest ascent/descent
653-654 surface 624 tangent plane 632 functions of many variables 683-707 chain rule 688 derivative, mixed 684 directional derivative 692, 693 envelope 702, 703,707 errors 685
gradient vector 688, 689 higher derivatives 684 implicit differentiation 686 incremental approximation
683, 684 Lagrange multipliers 699-701,
706
level surface 696 material derivative 690 normal to surface 690
partial derivatives 683 restricted stationary points
697-702 stationary points 696 tangent plane 691, 692
gamma function 372 gas, equation of state 687 gate (logic) 803
AND 804
EXOR 808
NAND 805
NOR 805
NOT 804
OR 804 Gauss-Seidel method 273
diagonal dominance 274 Gaussian elimination 263, 264
back substitution 263
echelon form 264
inverse matrix 265
pivots 264 generalized function 600 geometric distribution 891, 957
mean 891
variance 892 geometric sequence
(progression) 43 geometric series 43
common ratio 43
infinite 45
sum of 44,46 geometrical area 314
in polar coordinates 344 Gibbs' phenomenon 927 gradient (curve) 9, 61 gradient vector (grad) 659, 688
curvilinear coordinates 780
in cylindrical polars 778
identities 785
in spherical polars 781 graphs (see also curves) 7
gradient 61
sketching 108
slope 61—65 graph theory (networks) 814-841
bipartite graph 832
branch 821
circuits, electrical 821, 824-827, 838, 839
compatibility graph 836 complete graph 817, 833 connected graph 817, 820 corree 822 cutset 822, 929 cutset, fundamental 824 cutest, proper 823 cycle 820
degree of a vertex 817, 818 digraph (directed graph) 816 disconnected graph 817 edge 814
eulerian graph 821 Euler's theorem 832 face 832
frameworks 834-835 hamiltonian graph 821 handshaking lemma 818 labelled graph 818-820 link 822 loop 817 multigraph 817 node 815 path 820
path, shortest 816 planar graph 815, 831 regular graph 817, 820 signal flow graph 827-831 simple graph 817 spanning tree 822 subgraph 821
traffic signal phasing 835, 841 trail 820 tree 821
unlabclled graph 818-820 vertex 814 walk 820
weighted digraphs 828 gravitational field 755 Green's theorem 748 group velocity 436-437
Half-life 36 hamiltonian graph 821 handshaking lemma 818 harmonic forcing 399 harmonic function 21, 414
standard form 414 harmonic oscillation 413
phase diagram 483
phasor 443
harmonic oscillator 413-425 amplitude 414 angular frequency 415 damped 419 lead and lag 415 period 414
phase (difference) 415
wavelength 415
wave number 415 harmonics 562 Heavisidc unit function 14
Laplace transform of 519 histogram 903
homogeneous linear equations 271 (see also linear algebraic equations) Hooke's law 417 hyperbola 11
asymptotes 11
rectangular 626 hyperbolic functions 36,153
derivatives 88, 952
identities 37, 38, 951 (table)
inverse 38
inverse as logarithms 38 trigonometric functions, relation with 153 hypergeomctric distriburion 895 mean 895 variance 895, 957
Icosahedron 833 identity 5
impedance 446-451, 533-535
capacitor 447,533
complex 446-451
in frequency domain 446, 533
inductor 447, 533
parallel 448, 534
resistor 447, 533
series 448,534
in s-domain 533 implicit differentiation 654, 666, 686
implicit function 12 improper integral 328
convergence 330
divergence 330 impulse function 530, 599 (see
also delta function) impulsive input 544
increment 63
incremental approximation 115,
645,683 indefinite integral 324
table 953 identity matrix 170 index laws 4, 948 induction 843 inductor
impedance 533
phasor 446
complex impedance 447 inequality 5 infinite scries 128
convergence 129
divergence 129
geometric 43
partial sums 129
sum 128
Taylor series 130 inflection, point of 93, 238 inner product (see scalar
product) integer floor function 560 integers 4
sums of powers of 949 integrals 320-378 (see also antidcrivative; integration; double integral; line integral) and area 323, 333, 327 area, polar coordinates 345 area analogy 327,346 of complex functions 331 definite 323
differentiation of (variable
limits) 336 differentiation with respect
to parameter 374 even function 334 improper 328 indefinite 324 infinite 329
as limit of a sum 341-353 limits of integration, variable 336
numerical evaluation of 322,
339, 346, 355 odd function 334 rectangle rule 322, 347 Simpson's rule 355, 925 solid of revolution 343 square bracket notation 316
surface 765 symmetrical 335 table of integrals 953-954 trapezium rule 346, 347 variable limits 336 volume 765
integral equation 529, 559, 560 Volterra 559
integrand 324
integrating factor 407-410
integration 320-378 (see also integral; double integration) change of variable 362-366 of inverse function 370 partial fractions 366 by parts 368-373 reduction formulae 373 by substitution 356-366,378 of trigonometric products 362
interference 417, 456, 615
fringes 457 intersection (sets) 791 interval 5
infinite 5 inverse function 23-25
derivative of 94
integration of 370
reciprocal relations 23
reflection properry 24 inverse matrix 172, 190
Gaussian elimination 265 Inverse trigonometric functions 25
principal values 26 irrotational field 775 isocline 461,493 iterative methods {see
approximation)
Jacobi method (for linear equations) 274
Jacobian (double integration) 728
jump (discontinuity) 14
Kirchhoff laws 449, 824, 825 Kuratowski 833
Lagrange multipliers 667-672,
681,955 Laplace equation 785,786 Laplace transforms 505-561,
926-927, 955 (see also z-transform) convolution theorem 541, 726, 927
cosine function 507 definition 505 of derivatives 515 delay rule(second shift rule) 522
delta function 530 differential-delay equation 560 differential equations 516-519 differential equations,
variable coefficients 560 discrete systems 545 division by s 528 division rule 524 of Fourier scries 585 Heavisidc unit function 519 impedance, s-domain 530 impulse function 530 impulsive input 543 integral equations 529, 559,
560
inverse 505, 512 inverses, table of 955 multiplication by ekt 510 multiplication by /" 510 notation 506 of powers, t" 507 partial fractions 513 quiescent system 517 rules, list of 955 i'-domain 529 scale rule 508, 955 shift rules 510, 955 sifting 531 sine function 507 square wave 521 table of 513 and transfer function,
s-domain 535 and transfer function,
co-domain 540 Volterra integral equation 559 and z transform 548 lead and lag 415
least squares
estimates 914
method 638-640, 928 Leibniz's formula 123 level curve 625
normal to 657 level surface 696
normal to 696 l'Hopital's rule 136-138 light switches (Boolean
application) 811, 813 limit 65,72, 98,110,111
for derivative 65
important limits 72-76
left/right 111 limit cycle 497, 926
stability 499 line integrals 735-761
closed path 746
definition 736
evaluation 736
field, conservative 752, 753-759
field intensity 753
Green's theorem 748
non-conservative potential field 757
as an ordinary integral 736
parametric form 740-742
path 735
path dependence 736,738 path independence 744,
747-750 paths parallel to axes 743 of perfect differential 744-745 potential 755, 756 potential field 756 in two and three dimensions
739
work 750, 753, 755 linear algebraic equations
259-278 augmented matrix 263 back substitution 263 compatible 267 Cramer's rule 229 diagonal dominance 274 echelon form 264 elementary row operations 262 elimination 259 Gauss-Seidel numerical
method 273 Gaussian elimination 263, 265
geometrical interpretation 268 homogeneous 271 ill-conditioned 640 incompatible 267 Jaeobi numerical method 275 pivots 264
trivial, nontrivial solutions 271
linear dependence 185, 285 linear independence 286 linear oscillator 418-425 (see
also harmonic oscillator; oscillations) circuit model 418 damping 419 deadbeat420
free (natural) oscillations 419 overdamped (heavy damping)
420 resonance 423 transient 420 underdamped (weak damping) 420 link (graph theory) 822 In (see logarithm) logarithm 33-35
of complex number 157 derivative of 76 properties 34 Taylor series 131 logarithmic differentiation 92 logic gates (see Boolean algebra;
gate) 803 logic networks (see Boolean
algebra) 805 logistic equation 478, 845, 854-858, 860, 929 loop (of graph) 817
mass-centre 348, 350 mass-spring system 418 material derivative 690 Mathematica projects 920-930 matrices 161-176 (see eigenvalues; eigenvectors; matrix algebra; matrix, inverse) adjacency 837 adjoint (adjugate) 189, 190 augmented 263 characterstic equation 279
determinant of 175, 190 (see
also determinants) diagonal 170
diagonalization of 286-289 echelon 264 eigenvalues 281 eigenvectors 285 idempotent 301 identity 170 inverse 172 leading diagonal 169 lower triangular 274 non-singular 173 null 163 order 161
orthogonal 295-298, 923 positive-definite 295 powers of 171,289-292 quadratic form 292 rank 304 rectangular 162 row-stochastic singular 173 skew-symmetric 169 square 162 symmetric 169, 294 rrace of 301 transpose 168 unit 170
upper triangular 274 vector 162, 169 zero 163
matrix algebra 161-178 (see also matrices; matrix, inverse)
associative law 164
Caylcy-Hamilton theorem 302
conformable for
multiplication 165
difference 163
elementary row operations 262 equality 162
linear equations 259-278 multiplication 165 multiplication by a constant 163
multiplication on left/right 167
postmultiplication 167 prcmultiplication 167 row-on-column operation 165 sum 163
summation notation 166
matrix, inverse 172, 189, 923 (see matrix; matrix algebra)
by Gaussian elimination
of a product 174
rule for 2 X 2 173
rule for 3 x 3 175 maximum/m i n i mum
local 103
N variables 696
one variable 102
one variable, classification 104
restricted 107, 670, 697, 699 (see also Lagrange multipliers)
two variables 635-638
two variables, classification 637 mean (expected value,
expectation) 889, 897 median 906 mode 906 modulus 6
moment (see also force)
about an axis 253
of force 251, 255
vector 252 moment of inertia 348, 350-352
cone 377
disc 352, 377
rectangle 354
sphere 377
triangle 351 moment of momentum 258 mortgage 844 moving average 620 multigraph 817 mutually exclusive events 869
nabla (gradient) 659
negation (Boolean algebra) 804
negative binomial (Pascal)
distribution 894, 957 mean 894, 957 variance 894, 957 Newton cooling 412 Newton's method 116-119 nodal analysis (circuits) 824 node (graph theory) 815 node (phase plane) 488,494 nonlinear differential equations (see differential equations, nonlinear)
normal
to curve 238, 657, 658
to plane 232
to surface 632, 690 normal coordinates 299 normal distribution 898-900
standard normal curve 899
standardized 899,957
table 958 number line 5 number 3
complex (see complex numbers)
exponent (index) 4
exponent rules 4
index laws 948
infinity sign 4
integer 4
irrational 4
modulus 6
powers 4
rational 4
real 3
recurring decimal 4, 46 set notations 790 numerical methods (see approximation)
Ohm's law 824 operator 66 ordered pair 144 ordinate 6
origin of coordinates 6 orthogonal matrix 295-298, 923
rotation of axes 296 orthogonal systems
of coordinates 675
of curves 654,928 oscillations
addition 417, 454
beats 431-437
compound 431
damped 419
dcadbeat420,488
forced 420
harmonic 413, 427
longitudinal 298
overdamped (heavy damping) 420
transients 420
underdamped (light damping) 420
oscillator, linear 419-425 outcome 866 outlier 907
parabola 12
paraboloidal coordinates 785 parallelepiped
volume (determinant) 257
volume (vector) 251 parallelogram
area (determinant) 734
area (vector) 248 parallelogram rule
complex numbers 145
vector addition 201 parameter (statistics) 903 parametric equations of a curve
95,664 Parseval identity 585, 608 partial derivative 627 (see
functions of N variables)
higher 629
mixed 630
second 629 partial differentiation 623-705 partial fractions 39-42
in integration 366
and Laplace transforms 513
rules 40
and z-transforms 554 partial sum 129 Pascal distribution 894, 957 Pascal's triangle 52, 949 Path, phase (see line integral;
phase plane) path (graph theory) 820 pdf (probability density
function) 895 pendulum 71,394,425,489-491,
648,960
perfect differential form 471-473
line integrals 744 period 9,22, 414 period doubling 856 periodic functions 22 (see also harmonic functions; Fourier series) amplitude 22, 414 and Fourier series 563, 567
angular frequency 22, 415
frequency 22,414
integrals of 564
lead/lag 415
mean value 328
period 22, 414
phase 22,415
phase difference 415
spectrum 577
wavelength 19,415
wave number 428 permutations 46-51, 949
circular 49 perspective 243 phase (angle) 22,415 phase difference 415 phase plane 483, 488,491 (see also differential equations, nonlinear)
equilibrium points 484, 488, 493
general 491-497
limit cycle 497, 926
numerical method 499
path direction 484,488, 493
phase diagram 483
phase path 484, 488,493 phase velocity 429 phasors 442-459
addition 444
algebra of 444
Argand diagram 443
capacitor 447
complex amplitude 453-454 complex impedance 446-451 definition 443 of derivative 444 diagram 445
frequency domain 447, 451
harmonic oscillation 443
inductor 447
of integrals 444
interference 456
oscillations 453
resistor 447
time domain 447
transfer function 451
and waves 453 pitch 562 pivot 264
planar graph 815, 831 plane
cartesian equation 208
normal vector 231 tangent 632, 691 vector equation of 207, 232 Poisson distribution 892, 957 mean 893 variance 893
approximation to binomial distribution 893 polar angle 17 polar coordinates 28-30
complex numbers 146
cylindrical 704, 777
in double integration 721
geometrical area in 344
motion in 2(4-216
spherical 780 polygon
regular, area 951 polynomial 40
derivative 126
Taylor 125 population (statistics) 903
estimated mean 906 population problems 36, 393,
478, 491,560, 643 position vector 206
derivative of 213 positive definite matrix 295 potential 755
energy 71,755
field 756
single-valued 670 probability 865-883 addition law 871
axioms 870 Bayes' theorem 880 conditional 875-877 event 866 frequency 872
mutually exclusive event 869 and sets 868 total 879
Venn diagrams 868 probability distributions
884-902, 957 Bernoulli trials 887 binomial 887-888 continuous 895-900 counting method cumulative distribution
function (cdf) 895 density function (pdf) discrete 884-895
expected value 889 exponential 897 geometric 891 hypergeometric 895 independent event 877 mean 889, 897 negative binomial 894 normal 898
normal, standardized 899 Pascal 894 Poisson 892
probability density function
(pdf) 895 relative frequency 865 sample space 866 standard deviation 890 table 957 trial 865 uniform 901 variance 890, 897 product rule (differentiation) 83,
101
quadratic equation 140, 948 quadratic form 292
positive definite 295 quartiles 906
quotient rule (differentiation) 85, 101
radian 16, 949
radiation problems 613-618 (see diffraction, interference) radioactive decay 393 random sample 903, 910 random variable 884-902 (sec also probability distributions) continuous 895 discrete 884
mean (expected value) 889, 897
probability distribution (function) 885
standard deviation 890
variance 890, 891, 897 random walk 860 rank (of a matrix) 304 rational function 40 Rayleigh's theorem 607
reciprocal rule (differentiation) 85
rectangle rule (integration) 322, 347
recurrence relation 464 (see also
difference equations) recurring decimal 2, 46 red shift 438 reduction formula 373 regression 913—915
controlled variable 913
least squares estimate 914
linear model 914
line 915
response 913
scatter diagram 913
straight line fit 914
unbiased estimators 916 relaxation oscillation 499 repeated integral 709-717 {see double integration)
separable 724 resistor
impedance 533
phasor 447
complex impedance 447 resonance 423
restricted stationary values 106,
667,697 resultant of forces 235 root mean square (mis) 328 rotation of axes (see also axes)
223,226 row operations, elementary 262
saddle (phase plane) 484, 494 saddle (surface) 625, 635-638, 927
sample 903
mean 905, 910
standard error of mean 910
variance 911 sample space 866 (see also events)
countable 866
discrete 866
elements of 866
event 866
exhaustive 869
partitioning of 870
Venn diagram 868
sampling distribution 908 sawtooth wave 584, 927 scalar 219 scalar function 659 scalar (dot, inner) product 218-240 (see also vectors)
angle between vectors 221
of basis vectors 222
invariance 248
perpendicular vectors 222 scalar triple product 249
cyclic order 250 scale rule (Laplace transform)
508, 955 scatter diagram 913 separable differential equations
466-469 separation of variables 466-469 separarrix 491 sequence 43,129, 845
of partial sums 129 series (see Fourier scries;
geometric series; infinite series; Taylor series) sets 789-800
associative laws 793
binary 789
cardinality 798
cartesian product 800
commutative law 793
complement 791
complementary laws 794
de Morgan's laws 795
difference 794
disjoint 791
distributive law 794
duality 800
elements 789
empty 791
equality 790
finite 790
identity laws 794
infinite 790
intersection 791
number sets 790
ordered pairs 800
proper subset 792
subset 792
union 791
universal 791
Venn diagram 792 sgn (signum) function 15, 920
shah function 605
shift rule (Laplace transform)
510,955 shoulder (surface) 635 sidebands 585, 598 sifting function 531 signal 545, 589 signal energy 607 signal flow graphs 827-831
block diagram 827
cycle 830
edges in series 829 feedback 827, 828 loop 830
multiple edges 829 stem 830
weighted digraph 828 signed area 314, 320 signum (sgn) function 15, 920 simple harmonic motion (see harmonic oscillator) Simpson's rule 355, 925 sine function 594 sine function 18 (see also
trigonometric functions)
antiderivative 312
derivative 75
exponential form 150
rectified 582
Taylor scries 130 sine rule 950
sine transform (see cosine/sine
transform) singular matrix 173 singular solutions (of differential
equations) 468 sinh function 37
Taylor series 131 sinusoid 22
slope 9, 61-65, 66 (see curve; functions of two variables; straight line) solenoidal field 775 solid of revolution 343
surface area integral 343
volume integral 343 spectral density 588 spectrum (Fourier series) 577 speed 98 sphere
surface area 951
volume 951 spherical polar coordinates 780
spiral (curve) 28
archimidean 57
equiangular 57 spiral (phase plane) 487, 494 square root 4 standard deviation 890 standard error 910 stationary points (see also
maximum/minimum)
N variables 696
one variable 103
one variable, classification 104
restricted 107, 667,697
two variables 635
two variables, classification 637
statistic 903 statistical inference 903 statistics 903-916
bins 904
box plot 906
cells 904
central limit theorem 911
controlled variable 913
estimate 906, 909
estimator 909,911
frequency polygon 903
histogram 903
interquartile range 907
least squares estimate 914
mean 905
median 906
mode 906
outlier 907
parameter 903
population 903
quartiles 906
regression 913-915
regression estimator 915
regression line 915
response variable 913
sample mean 905, 910
sample variance 911
sampling distribution 908
scatter diagram 913
standard error 910
statistic 903
variates 903
whisker 907 steepest ascent/descent 654 stem 830
stiffness of spring 298, 417 Hooke's law 417
Stokes's theorem 784 straight line 8
cartesian form in rhree
dimensions 235 cartesian form in two
dimensions 8, 230 determinant equation of 191 direction ratios 230 gradient 9
parametric form 235 perpendicular 9 slope 9
vector equation 235 strange attractor 858 streamline 706 subgraph 821 subset 792
substitution, method of
(integration) 356-366, 378
summation sign 43 sums of integer powers 949 superposition principle 399 surface 625, 690
area 767
cone 625
contour map 625, 927 hemisphere 625 integral 765
maximum/minimum 635, 637 normal to 632, 690 parametric form 768 saddle 625, 635, 637,927 shoulder 635 stationary points 635 stationary points classification 637
tangent plane 632, 691 switches
light 811, 813
in parallel 810
in series 810
truth tables 810, 811 switching circuits (Boolean
algebra) 809 switching function 713
tangent (to a curve) 65 equation 66 vector 212, 238
tangent plane 632, 691
Taylor polynomial 125, 921, 922 Taylor series 124—138
binomial series 131
composite functions 132
general point, about 134
large variable 134
polynomial approximation 125
table 130 tetrahedron 773, 785, 923 thermodynamic equations 687 top-hat function 593
Fourier transform 593 rorquc 253 torus 768
total derivative 665 total probability 879 total waiting time 837 traffic signals 835-837, 841
compatibility graph 836
phasing 836
subgraph 836
total waiting time 837 trail 820
transfer admittance 452 transfer function 535, 536, 549 frequency domain 451 5-domain 535 transfer impedance 452 transform (see Laplace
transform; z-transform;
discrete systems; Fourier
transform; cosine/sine
transform) transient 421 translated function 13 transpose 168 trapezium rule 346, 347 travelling waves 427, 430, 434,
454
complex amplitude 454 tree (graph theory) 821
spanning 822 trefoil knot 923 trial 865
Bernoulli 887 triangle function 605
Fourier transform 605 triangle rule 200 (vectors) triangle, vector area 923 trigonometric functions 17-21
derivatives 18
exponential form 150
-
identities 20,155, 566,	multiplication by a scalar 200	solenoidal 772,775
949-950(table)	normal to curve 680	Stokes's theorem 781-784
integrals of products 362	normal to plane 231	surface area 767
inverse 25	normal to a surface 632,690	surface integral 765
Taylor series 130	parallel 199	triple integral 769
truth tables 803 (Boolean	parallelogram rule 201	volume integral 765, 769
algebra)	perpendicular	vorticity 782
for gates 803	plane equation 208, 231	vector (cross) product 244-258
inverse method 808	position 206	direction of 246
for switches 809	and relative velocity 204	invariance 248
	right/left-handed system of	rules 245
gm	198	of unit vectors 245
ISC	row 162, 169	vector space 285
uniform distribution 901	rules of vector algebra	base vectors 285
union (of sets) 791	199-202	vector triple product 255
unit step function 14	scalar (dot, inner) product	velocity 67, 212, 213
units (SI) 959-960	220	angular
unit vector 211, 223	scalar triple product 249	polar components 216
universal set 791	straight line 209, 230, 234	relative 204
	subtraction 200	Venn diagram 792, 868
mm	sum of 181	vertex 814
w	tangent to curve 213,238	vibrations (see oscillations)
valency 816	triangle rule 200	volume
van der Pol equation 499, 926	unit 211,223	of cone 342, 951
variable, random (see random	vector product 244 (see vector	ellipsoid 353
variable)	product)	integral 769
variable, dependent/independent	vector triple product 255	parallelepiped 251, 257
12	and velocity (see velocity) 213,	of solid of revolution 343
variance 890, 897	216	table 951
variance, sample 911	vector fields 762-786 (see also	tetrahedron 785
variate 793	curl; divergence;	
vectors 193-258 (see also axes;	gradient)	
vector field)	cylindrical polar coordinates	II
acceleration 213	777	walk (graph theory) 820
addition of 200	curl 773-777	walk, random 860
angle between 220	curvilinear coordinates	water clock 354
basis 210	779-781	wave (see also diffraction;
column 162,169	divergence 764	interference;
components 199	divergence theorem 770-773	oscillations)
coplanar 203, 251	field lines 762-764	antinode 427
cross product (see vector	fluid flow 772,774,775	attenuating 629
product)	flux 770	beats 431,432,434
and curvature 238	gradient 780	carrier 432, 585, 598
differentation 212-214	identities 785	complex amplitude 453-454
directed line segment 198	integral curves 762	compound oscillation 431
displacement 193,197	irrotational 775	diffraction 417
dot product (see scalar	Laplace's equation 786	dispersive 436
product)	orthogonal coordinates	Doppler effect 437
equality 199	[general)	equation 631
gradient 659-661, 688	paraboloidal coordinates 785	frequency modulation 435
(see also gradient)	scale factor 778	group velocity 436
invariance 248	spherical polar coordinates	intensity 455
magnitude (length) 199	780	interference 417
modulation 432, 435 node 427 number 415 phase velocity 429 plane 429, 430 progressive 428 sinusoidal 427 standing 427 stationary 427 train 429
travelling 428,430,434 wavelength 19,415
wave number 428 wave packets 432 wave train 429
Whcatstone bridge 122, 449, 451 whisker 907 work 341,750-752
z-transform (see also discrete systems) 548-561 convolution theorem 552
complex plane 522—556 definition 549 delay circuit 547 difference equations 556 differentiation analogue 561 discrete signal 561 inverse 549 linear system 555 poles of 554
stability of discrete system 554 time-delay rule 561 transfer function 549
feedback on the previous edition:
'to this edition:
This textbook offers an accessible and comprehensive grounding in many of the mathematical techniques required in the early stages of an engineering or science degree and also for the routine methods needed by first and second year mathematics students.'
ENGINEERING DESIGNER, 2003
'The authors do not attempt to dodge theoretical hurdles. They are careful to explain many of the less intuitive properties of functions and to highlight generalizations without becoming over abstract... There are significant changes in content in the opening chapter, where the foundation material has been expanded usefully.'
TIMES HIGHER EDUCATION SUPPLEMENT, NOVEMBER 2002
'Thoroughly recommended.'
ZENTRALBLATT MATH, 2002
Mathematical Techniques covers all the topics required by a science, engineering or mathematics student in the early stages of their degree. Assuming a minimum of prior knowledge, basic subjects are covered from scratch or topics are revised anew, giving a firm foundation for the topics that follow.
With a huge array of end of chapter probLems, and new self-check questions, the fourth edition of Mathematical Techniques provides extensive opportunities for students to build their confidence in the best way possible: by using the maths for themselves.
• Short, modular chapters, a large index, and careful cross-referencing make the book flexible enough to be used on a wide variety of courses.
• Over 500 fully-worked Examples show how the theory is applied and offer valuable guidance for the reader when tackling the problems.
• Self-check questions and over 2000 end of chapter problems provide extensive opportunities for students to practise using the concepts presented.
• Emphasis on techniques and applications helps readers to move through the subject without being hampered by formal analytic proofs.
• A series of Projects at the end of the book encourages students to use mathematical software to further exploit and ilLustrate the concepts covered.
Each chapter opens with a new introduction, which explains the content and aim of the chapter, and places it in context for the student. The whole text has been reviewed, and colour introduced, in order to improve clarity and to facilitate navigation through the text.
New self-check questions appearat the end of most Sections. These supplement the end of chapter problems, and provide the opportunity to check progress.
A new Appendix covers Dimensional Anatysis and Units.
Topics revised in this edition include conic
ctions, linear dependence, integration, nonlinear differential equations, stationary values, vector fields and Stokes's theorem, and difference equations.
Cover illustration: Simon Witter
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