Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/266

 When $$t = 0$$, we have from (D), (G), (J), (K), (L)),

and so on. Substituting these results in (E), we get

To get $$f(x + h, y + k)$$, replace $$t$$ by 1 in (66), giving Taylor's Theorem for a function of two independent variables,

which is the required expansion in powers of $$h$$ and $$k$$. Evidently (67) is also adapted to the expansion of $$f(x + h, y + k)$$ in powers of $$x$$ and $$y$$ by simply interchanging $$x$$ with $$h$$ and $$y$$ with $$k$$. Thus

Similarly, for three variables we shall find

and so on for any number of variables.