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.

~ 53. 54.

Higher partial derived functions.

95

which must be denoted by -y, and likewise when is differenoyOx' a)y
2f
tiated with regard to x, -— y The two are defined through the
original function by the equations
f(x+-Ax, y+Ay) -f(x, y+Ay) f(x-A x, y - f(, y)
a2. Ax Ax
2- - Lim Lim
aYax 0y=O Jx=O Ay
f(x+Ax,y+Ay) - f(x+Ax,y) f(x, y Ay) -f(x, y)
at2 Lim Lim A y Ay
axay x-o y=o A x
The expressions on the right are identically equal to:
f(x + A x,y + Ay)- -f ( x, y + Ay)- f(x A,y) f(x, y) (Ax, A)
A XAy
and differ from each other only in this, that in the first case the
limiting value has to be formed by making A x first converge to zero
and then A y, on the other hand in the second case in reversed order
first Ay becomes zero then Ax; the question is, must these limiting
values be identical? We assert: They are identical always at a point,
in whose neighbourhood a and (or a and f ) are
cniuu o ax a (or again a xayd TX ay ar
Ax ~yx d y axay/
continuous functions of both variables x and y; without implying that
this is at the same time the necessary condition. For, in consequence
of our hypotheses we can apply the Theorem of the Mean Value to
the function
(x, y) - f(x, y + Ay) - f(x, y)
in which we first consider y and A y as constant, and x as variable
OTT(x OA x, y),
nT(x +Ax, y) - T(x, y) = ( + ax.x,
or in full
{ f(x+ A x, Ay)-fx+ Ax )} - fx,) (x,y +Ay)-ft(x,y)}
- Ax t f(x+ Ox,y Ay) af(x+ Ax,y)}
ox 8x
Accordingly
ft(x+OAx,y+Ay) a f(x+OAx,y)
(A xAy)^ = ax ax 2 f(x OAx, y+ -Ay)
";'/Ay ayax
For the theorem of the mean value, by hypothesis, likewise holds for
a f on / f bo cn
the function at. If now- / be a continuous function of both
vaix blsitf OA X, y + Ay) yieldsA
variables in the neighbourhood of x, y, f+ y - ) yield
the same value, whether Ax first vanishes and then Ay, or first Ay
and then Ax, i.e. I presents the same value independently of the
order Ax -=0, A y = 0. Therefore in this case the limiting values
a02 a2f
and ax are identical.
~yax axay