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.

90

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

But this not being so, a finite region + Oh, + k can be assigned
at the point x1, y1, for which
abs [f(x1+ ~ Oh, yJ ~+ -) - f(x,, y)] < d.
This finite region includes also the points x1 --, y - s' (for E and
E' converge to zero, h and k have fixed finite values) and therefore
the same assignable region is also sufficient for each of these points
for satisfying the inequality, so that therefore what was assumed is in
contradiction with the condition of continuity.
53. The first differential quotients of the function at a point, in
whose neighbourhood it is continuous, can be formed in various ways:
If we first leave y unaltered, while x increases or decreases by Ax,
and denote the corresponding change of z by Age, then the quotient
of differences is
AxZ f(x + Ax, y) - f(x, y)
Ax + Ax
We assume that this approximates, when Ax converges to zero, to a
determinate limiting value, as well for the + as for the -- sign, but
not necessarily the same for both. It is denoted after Jacobi by
z or Of and called the partial derived of z with regard to x
progressive or regressive as the case may be, thus:
af a= Z Lim f(x +Ax, y- f(x) for o.
ex x - + A x
The partial derived of S with regard to y secondly is got
in the same way, x remaining unchanged:
Of Z Lia f(x, y t Ay)- f( x, y)fr
- ay Lim for Ay ==- 0.
ay ~y +Ay
Obviously we have here also the proposition: If the progressive
partial derived with regard to x or with regard to y be identical with
the regressive one, the Theorem of the Mean Value holds:
f(x + h, y) - f(x, y)= Zf' (x + Oh, y),
f(x, y + k) - f(x, y) = kf' (x, y + 7 k),
0 and i will be respectively dependent on y and x.
But if x be changed by Ax and at the same time y by Ay, the
ratio Ay: Ax remaining quite arbitrary but finite, the increase is
Az- f(x + Ax, y + Ay) -f(x, y).
Here also the question arises: What is the limiting value to
Az Az
which A- or - tends, when Ax and Ay converge to zero in any
an w t r a ys r a
manner in which their ratio always retains a finite limiting value
dx'