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.

~ 55-66.

Taylor's series.

99

~3 f a3 f a3 f s3 f9
d3z~ — x3-JX3 _ 3 dx2dy +3 + dx dy2+ dy3
+ 2 (IX (1zf d x d — (d-x dy + dy + dx)+ y d y clyd + af 63x + f d 3yd.
But if x and y are both to be taken as independent variables, this
expression reduces to its first four terms. It is seen that these terms
occur with binomial coefficients and that in general for independent
variables
— n
ding — ',k ~x__ _* f dx-' d y, (0 = 1).
For, if we form the total differential of this equation, we find:
~dn+n1,n-k+1 f1 d=x-k dyy + dy Z nk n Zx dx-yk+
Except the first term of the first sum and the last of the second,
each. term occurs twice, only with a different binomial factor, so that
ag+la f a+lf
d" + -- dx"+' + (n2 + 2) a dx' dy -- ~ ~ ~
a xn+1 axnay
+ a"fl r+l fd yn+
+ (flk + hx-1) kdxn'+-l dyk +,. ay +
but since nk + T_-1 = ( + l) )k, we have
k —n+t [n+lf
dnl (t + 1)k k+_ dx+'- k dyk, Q. E. D.
k=-O
There is no difficulty in extending these investigations to explicit functions with more than two independent variables, when once
we have defined for them in a perfectly analogous manner what is
meant by the continuity of the partial and total differential.')
56. The knowledge of the partial derived of a function with
more independent variables than one, leads likewise, as Lagrange has
shown, to a calculation of the function by an infinite series of powers.
To find the value of -- f (x + h, y + k), when the values of the
function and of all its partial derived functions at a point x, y are
known, let us form the expression
1) F (t) =- f(x + th, y + tk) = f(x', y').
This will be a continuous function of t for arbitrary values of h and
k, only if f is a continuous function of the two variables within the
region determined by h and k. Now if F (t) can be expanded by
Mac Laurin's series, that is, if
~) Theorems concerning functions with more than one variable were first
systematically developed by Euler: Instit. calcul. diff, Pars 1, 7.
7*