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.

100

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

2) F (t) =-F () + t y (0) + t F"(0) +.. Fn (t),
we get for t= 1 the value:
3) F (1)= F(O) (-+ 17 F (0) + a F" (0) + * * F (6).
Now on the hypothesis that f and its partial derived functions are continuous in both variables, we have for every value of h and k the total
derived of F with regard to t:
af(x', y') dx' I f(x', y') dy'
i' (t)- a x' dt +- ' ct
or because: x'= x + ht, y' = y +- kt, therefore
df -f af O a f dx' dy'
x' ~ x y'y' d y ' dt dt
F' (t) - h fx ) + k f x' and F' (0) = h xf(y) + k Tf(xy)
Further we have
C" f(x', y' ) ( k2 '2f (x', y')
F" (t) 2= h " + 2 hix 2y' + 2 '/(,')
aX2 2x o2 a?
therefore
F" (0) - 2 + 2 h ^ f (x, Y) 9 d 2f(x, y)
x y ^^2 ' x2y + k y2
and so on, in general:
Fn (t) n --- np h-P alP f X ').
p=O n-2 Ap
If we substitute these values in equation 3) we find
F(1)=f(x+h,y+k) = f (x, ) + |h { + }7
df f+ j}.. ] pnc k 'f'~(x+Gh,y+Oek)
h I 2 a9 t + 2 jk d2 f- + k2 2n +1 n-P k_
+ 2 ax2 3axa I, 1Ly Ln ay2x-t~ tn xn-p ayp
This expression is the Theorem of the Mean Value in its most
general form for a function with two variables. It leads to an infinite series proceeding by powers of h and k, whenever the remainder
converges to zero as the values of n increase arbitrarily. A special
case in which this occurs is when the partial derived functions have
the property of remaining finite in the domain assigned by ih and Ik
when n becomes infinite. If they have not this property, the remainder
may indeed still converge to zero, though, as the determination of
the limit becomes difficult, we require other criteria to decide by.