Siberian Mathematical Journal

, Volume 54, Issue 3, pp 566–573

On Taylor’s formula for functions of several variables

Article

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.

Keywords

Taylor formula rectifiable curve remainder functions of class Cr

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References

1. 1.
Reshetnyak Yu. G., “Remark on the integral representation of differentiable functions of several variables,” Sibirsk. Mat. Zh., 25, No. 5, 198–200 (1984).
2. 2.
Kondurar V., “Sur l’intégrale de Stieltjes,” Mat. Sb., 44, No. 2, 361–366 (1937).Google Scholar
3. 3.
Reshetnyak Yu. G., “On parallel translation along an irregular curve in a principal bundle,” Siberian Math. J., 13, No. 5, 739–755 (1972).
4. 4.
Reshetnyak Yu. G., “The notion of lift of an irregular path in fiber bundles and its applications,” Siberian Math. J., 16, No. 3, 453–460 (1975).
5. 5.
Ciarlet Ph. G. and Mardare C., “Recovery of a manifold with boundary and its continuity as a function of its metric tensor,” J. Math. Pure Appl., 83, No. 7, 811–843 (2004). 