An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

96

Functions of more than one independent variable. Bk. I. ch. IX.

The only remark still to be made is that the value of o - or
ayax
can also increase beyond all limits; though under the circumstances
assumed, namely that the theorem of the mean value is to continue to
hold, this can only occur by 4p becoming determinately infinite however
we may approach this point; the limiting values then remain equal to
one another, + oo or -- o.
We can now further conclude, that under corresponding circumstances the order of differentiation is indifferent also for higher partial
derived functions. For if
a2f '2f
Os/' fX y
ayax x ay
then it follows by differentiation ex. gr. with regard to x that:
33f a3f
f fa Y - x. ay h
If then we put f = p and the theorem just proved holds for the
function p, that is to say, if not only p- g but also
____/ a3f
ayax cayaxax
be a continuous function of both variables, we have:
I9p a'23 a3f /3f' a; 3 '
_Ad. av,i. e a^ a ^ z = qy =-, - -. Q.E D.
excy oyax ' ' axayax cy x Cx2cy
55. By the help of the higher partial derived functions the higher
total differential quotients are expressed as follows: In the function
dz __ af(x,y) a f(x,y) dy
dx Ax ay dx'
which depends on the two variables x and y, let x increase by A x,
dz
y by Ay, then the limiting value of the quotient of differences A dx
Ax
which we shall have to denote by dX2 when Ax vanishes, is to be
calculated from the form:
f(x +- x, y+ ) _ 8f(xy)
d2 Z ax ax
_ Lir
d x2 Ax
j af(x +Ax, y +A A) f(x,y) dy
dCyyay a Afx, y) dx.
+ -y Lim |- Y- -+ a ij.Lim dx
dx- Ax A a?) zAx
The first limiting value on the right is by the previous propositions the total derived function of with regard to x, therefore
is equal to x +f c dy; likewise the second, the total derived
d Cx2 ' y aox dxI