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
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