On Taylor’s formula for functions of several variables

Abstract

Elementary courses in mathematical analysis often mention some trick that is used to construct the remainder of Taylor’s formula in integral form. The trick is based on the fact that, differentiating the difference $$f(x) - f(t) - f'(t)\frac{{(x - t)}} {{1!}} - \cdots - f^{(r - 1)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $$ between the function and its degree r − 1 Taylor polynomial at t with respect to t, we obtain $$ - f^{(r)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $$ , so that all derivatives of orders below r disappear. The author observed previously a similar effect for functions of several variables. Differentiating the difference between the function and its degree r − 1 Taylor polynomial at t with respect to its components, we are left with terms involving only order r derivatives. We apply this fact here to estimate the remainder of Taylor’s formula for functions of several variables along a rectifiable curve.