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.

94

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

in such a case there can no longer be any such thing as a fixed
plane; the first partial derived functions are indeterminate. Let the
coordinates of the points be respectively x, y, z; x + A x, y + Al y,
+ -Ai; x + A2x, y +- 2y, z + A2z, so that
- f(x, y), + Atz - f (x + Ax, y -+ AM^),
z + A2 = f(x + A2, +- A2Y).
The equation of a plane through these points, denoting its current
coordinates by A, q, g, is:
- = A ( - x) + B( - ).
A, A2y A2z A1,
A - AlzA2y-A2z A1y _ A1,x A2x: A2X Ar
lxA2y- A2xAly A2 Y A_ y
A2x AIx
A2Z Az X
B A2ZAA X - A^ZA2 _ Aa2x i x
A1 xA2y- X Al y A2Y _A y
A2x AIx
Now if Atx and Ay, as well as A2x and A2y converge to zero,
while the limiting values of their ratios are denoted respectively by
(d) ' (d-),' we have, provided the theorem of the total differential
holds,
A1 aOf af (dy) A2z O_ f (dy
IX x ~ y \x /' 2-x ax ay \dx,
therefore: A - B =O f And it follows conversely from these
aro' a
equations: if A and B in every limiting process take these values, then
the Theorem of the Total Differential holds.
54. We now proceed to form the partial derived functions of
higher order according to the rules for functions with one variable.
By a- we denote the function which is found by taking the derived
of U with regard to x, and can again define it through the original
function by means of the equation
f- Lim f(x + 2Ax, y) - 2f(x x, ) + f(x, y)
x- Ax -X2
for Ax - 0. Similarly
Of
02f -- y = Lim f(x, ) -, y + 2 A y )2 + f(x, y)
3y2 ay ~Jy=O Ay2
By further differentiations the deriveds anf and f are obtained. But
a X ayn
there can also be formed what are called mixed derived functions: for
if the function if be differentiated with regard to y, a function is found