Differential Calculus

THIS book is intended to provide the university student in the physical sciences with information about the differential calculus which he is likely to need. The techniques described are presented with due regard for their theoretical basis; but the empha

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DIF-FERENTIAL CALCULUS

TELEPEN

LIBRARY OF MATHEMATICS edited by

WALTER LEDERMANN n.5c., Ph.D., F.R.S.lId., Seal« Lecturer in Mathematics, University of Manchester, Manchester

Linear Equations

P. M. Cohn

Sequences and Series

J. A. Green

Differential Calculus

P. J. Hilton

Elementary Differential Equations and Operators

G. E. H. Reuter

Partial Derivatives

P. J. Hilton

Complex Numbers

W. Ledermann

Principles of Dynamics

M. B. Glauert

Electrical and Mechanical Oscillations

D. S. Jones

Vibrating Systems

R. F. Chisnel1

Vibrating Strings

D. R. Bland

Fourier Series

I. N. Sneddon

Solutions of Laplace's Equation

D. R. Bland

Solid Geometry

P. M. Cohn

Numerical Approximation

B. R. Morton

Integral Calculus

W. Ledermann

Sets and Groups

J. A. Green

Differential Geometry

K. L. Wardle

Probability Theory

A. M. Arthurs

Multiple Integrals

W. Ledermann

DIFFERENTIAL CALCULUS BY

P.

LONDON:

J. HILTON

Routledge & Kegan Paul Ltd

NEW YORK:

Dover Publications Inc.

First published I958 in Great Britain by Routledge r.

give the explicit formula for dy, namely dx dy dy dz Theorem 2.5. dx = dz dx'

Let y-j(z) and z=g(x); we wish to prove that lim f(g(a+h)~-f(g(a)) j'(g(a)) .g'(a). 11-+0

Now, by definition, lim g(a+~)-g(a) g'(a); this is precisely h~O

equivalent to the statement g(a+h) =g(a) +h(g'(a) +e(h)) (2.6) where e(h) is a function of h which tends to 0 as h~o. Put g(a)=b, g(a+h)=b+k. Then k~o as h~o. Moreover, f(b+k)=f(b) +k(/'(b) +'TJ(k)), where 'TJ(k)~o as k~o. Thus f(b+k) - f(b) =k(/' (b) +1)(k)) = (g(a+h) -g(a)) (/' (b) +1)(k)) =h(g'(a) +e(h)) (/' (b) +1)(k)) =h(g'(a)j'(b) + fl,(h)) , where ,u{h) = e(h)j' (b) +g'(a)1)(k) +e(h) 1) (k) and thus tends to 0 r8

RULES FOR DIFFERENTIATING

as h-+o. We have proved that

J(b+kl-J(bL~g'(a}f'(b}

as

and this is precisely what was to be proved. The Leibniz notation makes this theorem look far more obvious than it really is. A proof may be given by invoking the h~o,

truly trivial statement

~= ~

:; and taking limits but there

are difficulties associated with possible zero values of lJz. There is no convenient generalization of Theorem 2.5 to higher derivatives. As an example we will show that dd2y is not XII

d2y d 2z equal to dz 2 dx s'

+

Example 2.4. d2y = d2y(dZ) s dy d2z. dx ll dz 2 dx dz dx" F

d2y

d (dY)

d (dY dZ)

or dx2= dx dx = dx dz· dx

_ d (dY ) dz dy d2z - dx dz . dx + dz' dx2' by Theorem 2·3· Again, by Theorem 2·5,

Y)

:iiz) !(iz);; =

(we replace y in

Y) = d2y dz and the formula 1S . dz dx'

d (ddz Theorem 2·5 by ddz . Thus dx

2

demonstrated. A most important special case of Theorem 2.5 is the following. Suppose, for a given range of values of x, the function J(x} is one-one. That is to say, suppose that, in this range, different values of x give rise to different values ofJ(x). Then x may be considered as a function of y over this range. Suppose, to be precise, that J(x) is a one-one function from the interval XO