A Course of Mathematics for Engineers and Scientists. Volume 4

359 Pages·1964·5.265 MB·English

by  Brian H. Chirgwin and Charles Plumpton (Auth.)

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A COURSE OF
MATHEMATICS
FOR ENGINEERS
AND SCIENTISTS
Volume 4
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Other titles of interest:
BALL: An Introduction to Real Analysis
CHIRGWIN & A Course of Mathematics for Engineers and
PLUMPTON: Scientists (6 vols)
GOODSTEIN: Fundamental Concepts of Mathematics
HOWSON: Mathematics for Electronic Technology
2nd Edition
PLUMPTON & Theoretical Mechanics for Sixth Forms
TOMKYS: 2nd (SI) Edition (2 vols)
PLUMPTON &
TOMKYS: Sixth Form Pure Mathematics (2 vols)
ROMAN: Some Modern Mathematics for Physicists and
other Outsiders (2 vols)
WOLSTENHOLME: Elementary Vectors 3rd (SI) Edition
A COURSE OF
M A T H E M A T I CS
FOR ENGINEERS
AND SCIENTISTS
Volume 4
BRIAN H. CHIRGWIN
AND
CHARLES PLUMPTON
DEPARTMENT OF MATHEMATICS
QUEEN MARY COLLEGE
MILE END ROAD, LONDON Ε. 1
PERGAMON PRESS
OXFORD NEW YORK PARIS
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First Edition 1964
Reprinted (with corrections) 1977
Reprinted 1978
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Printed in Great Britain by
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ISBN 0 08 009377 9
PREFACE
In this volume we continue the course of mathematics for under-
graduate students reading science and engineering at British and
Commonwealth Universities and colleges. The aim of this volume is
to generalise and develop the ideas and methods of earlier volumes
so that the student can appreciate and use the mathematical meth-
ods required in the more advanced parts of physics and engineering.
The elementary ideas of vector algebra are generalised and devel-
oped in two ways. First, in Chapter I is an account of vector analysis
and the differential and integral operations and theorems concerning
vectors. These ideas find their first generalisation in tensor analysis
and the transformation of coordinates, including orthogonal curvi-
linear coordinates. The second development, in Chapter V, is to
matrices, where the properties of arrays of elements, linear equa-
tions and quadratic forms are seen to be the generalisations of ele-
mentary algebra and, using Vector space', of familiar geometrical
ideas to η dimensions.
The solution of differential equations by series provides a very
general method for the solution of ordinary and some partial differ-
ential equations. A discussion of the properties of the solutions in
the light of the Sturm-Liouville theory introduces the conceptions
of eigenvalues and orthogonal functions, forming a link with ma-
trices. The Chapter on the special functions gives some of the better
known properties of Bessel, Legendre, Laguerre and Hermite func-
tions, which commonly occur in the solution of boundary and initial
value problems. These properties are linked with the series solutions
and orthogonality properties discussed in the preceding chapter.
The exercises and examples (taken mostly from examination pa-
pers) provide a number of applications to physical problems. We also
give a short bibliography where readers can find fuller discussions of
vii
viii PREFACE
many of the topics in this volume. These discussions consider with
more rigour and in greater detail many points which we have been
unable to include here, and also develop the treatments further.
We wish to express our thanks to the Senate of the University of
London, to the University of Oxford, and to the Syndics of the Cam-
bridge University Press for permission to use questions which have
been set in their examinations. We also wish to express our thanks
to our colleagues, and in particular to Professor V.C. A.Ferraro and
Dr. A.Mary Tropper, and to Mr. I. B. Perrott of Leeds University,
for their help, advice and comments on many occasions.
CHAPTER I
VECTOR ANALYSIS
1:1 Transformation of coordinates
Vectors and scalars were first introduced in Vol. II Chapter IV where
they were used mainly in geometry; subsequently, vectors were used in
connection with forces, velocities, accelerations, etc., in theoretical me-
chanics (Vol. III). We regarded a vector as a directed segment of a
straight line and developed vector algebra by defining products of differ-
ent kinds. We now extend these ideas so that they can be used in con-
nection with continuous systems such as electric, magnetic and gravi-
tational fields, the motion of fluids, conduction of heat, elasticity, etc.
Vector analysis is related to vector algebra in much the same way as
infinitesimal calculus is related to elementary algebra. Because we want
to use vector methods in connection with continuous systems, we must
unite the concepts of differentiation and integration with those of
vectors.
Since a vector includes the concept of direction in space, we must
always have a frame of reference against which to 'see' this direction.
Nevertheless many properties of a system must be expressed in rela-
tions which do not depend on the use of any particular frame of refer-
ence. Our first step in developing vector analysis, therefore, is to in-
corporate this independence of the frame of reference precisely and
explicitly in a modified definition of a vector.
We suppose that we have two rectangular frames of reference Οχ λχ%χζ
and Οξ1ξ2ξ3 which are both right-handed, and a vector a which has
components (αχ, α2, a3) and (at, α2, a3) in the respective frames. We let
the unit vectors e x, e2, e3 denote the directions of the axes of the frame
Οξ1ξ2^ referred to the axes 0 X^ XQ ·
The components of these three
unit vectors in the frame 0 x^ χ^ are
e e e
l (^11' hi' ^3l) > 2 (^12 ' ^22 ' ^32) ' 3 (^13 > ^23 ' ^33) ·
1
2 A COURSE OF MATHEMATICS
The first suffix of lt idenotes the axis in the x-fram3, 0xt, and the second
suffix denotes the axis in the ξ-frame, Οξ1 }which together enclose the
angle whose cosine is lt 1(i, j = 1,2, 3). By our definition in Vol. II of
the components of a vector as orthogonal projections on to the coor-
dinate axes we see that
ax = a · ex, a2 = a · e2, a3 = a · e3. (1.1)
a α A
Therefore
l — *Ί11 ~Γ" ^21^2 ~Γ" hi3
a
2 — ^12^1 ~f~ ^22^2 ~ί~ ^32% (1.2;
α a a a :
3 ~ ^13 l + ^232 + ^333
These can be summed up in the single relation
j = 1,2,3). (1.2a)
Since the position vector r of a point is a special case of a vector, the
coordinates of a point, xi (i = 1, 2, 3) in the #-frame and £j (j = 1,2,3)
in the |-frame, are also related by
ξι = Σ*ν**' (1-3)
k
(Henceforth all summations are to be taken from 1 to 3 over the suffix
indicated.)
It is convenient to set out the direction cosines in an array, called a
square matrix [Chapter V],
' ^11 ^12 ^13
L
hi hz hz
v^31 ^32 ^33 j
This array (matrix) has some very important properties.
(1) Since ef = e| = e| = 1, the squares of the elements in each col-
umn of L add up to 1, viz.,
j= 1.2,3) (1.4)
§1:1 VECTOR ANALYSIS 3
e a r
(2) Because the axes 0^2^s mutually perpendicular, the scalar
products vanish, viz.,
e e ee e e
2 ' 3 = 3 * l = l · 2 = °- (1-5)
Hence the scalar products of different columns of L are zero. We can
write (1.4) and (1.5) in the form
=ί
«4·«ί = Σ 'Λ ·ί (i,? = 1,2,3). (1.6)
where by is the Kronecker δ-symbol, which is defined by;
bt j= 0, if i Φ j; d t j= 1 if i = j.
(3) The matrix L stands for the array of the elements since it is
a square array we can combine these elements into a determinant,
detL, by the rules given in Vol. II Chapter III. Since the frame
Οξ1ξ2ξζ is right-handed
e e e e ee ee e
l ~ 2 ^ 3' 2 ~ 3 * l' 3 ~ l * 2 * (L7)
These relations imply that any element of det L is equal to its own co-factor,
and
detL = 1(^22^33 ^23^32) "f" ^21(^32^13 ~~~ ^12^33) H~ ^31(^12^23 ~~ ^22^13)
= l\ 1 + Zf 1 + ^31 " I ·
[If the frame Οξ1ξ2ζ^ is left-handed, the signs of the r.h. sides in (1.7)
are changed and det L = — 1.]
The axis Ox1 makes angles with the axes 0ξΐ 9 0|2, 0|3 whose cosines
are lj1 , li, 2^3 respectively. Therefore the unit vectors i x, i2, i3 directed
along the axes 0x±, Ox2, 0x3 respectively have components in the
frame Οξλξ2ξζ given by
*1 (^11» ^1 2 ' h 3) > h (^21 > ^2 2 > ^23) > *3 (^31 > ^3 2 ' ^33) >
i. e., the roms of L. [This result could be obtained by putting x k =(1,0,0)
for in eqn. (1.3), # Λ = (0, l,0)f or i2, etc.] By arguments similar to those
above we deduce
I W P = ii, = 1,2,3). (1.8)
ν
[The reader should note the difference between (1.6) and (1.8) in the
position of the summed suffix.]
4 A COURSE OF MATHEMATICS
For a vector a,
1,2,3) (1.9)
a relation which could also be obtained by solving eqns (1.2) using the
properties (1), (2), (3) of the direction cosines These results are
summed up in the following table.
Components in frame Components in frame
VVeeccttoorr
r (position vector) (fi.fa.fs)
a (a1 9α2, a3) («1. <*2> «s)
h (1,0,0) (^H> ^12» hs)
h (0,1,0) (^21» ^2 2> ^23)
eh (0,0,1) (^31> ^3 2» ^33)
e
i (^n» hi> ^31) (1,0,0)
e
2 (^1 2 > ^2 2 ' ^32) (0,1,0)
3 (^13> ^2 3> ^33) (0,0, 1)
Bearing these considerations in mind we now adopt a modified de-
finition of a vector as follows:
A vector is an entity having three components in any rectangular frame
of reference, the components in any two frames being related by the trans-
formation (1.2), or (1.9). (The definition of the sum of two vectors by
addition of components is not modified. The definition above is strictly
that of a cartesian vector.)
This definition implies that, if we are given that three quantities are
the components of a vector in one frame, eqns (1.2) give the components
of that vector in any other frame. On the other hand, many problems
provide sets of three numbers associated with a property such as angu-
lar velocity, magnetic field, etc., and with a frame of reference. If we
can show that these quantities are related by the transformation (1.2)
or (1.9), we have proved that they are the components of a vector. In
such cases we may therefore refer to angular velocity, magnetic field,
etc., as vectors. A scalar is a quantity which is not associated with
direction and which does not alter in value as a result of a change in the
frame of reference.