# Multi-Dimensional Taylor Series¶

date: | 2012/10/8 |
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Everyone knows how to write the Taylor series. Given a scalar function \(f(\xi)\), its Taylor series can be written as

where \(\xi_r\) is the centered point. There is few challenge in obtaining Eq. (1). However, if the function of interest has more than one variable, e.g., \(u(x_1, x_2, \ldots, x_N)\), Eq. (1) can no longer be used.

To obtain a formula for the Taylor series of a multi-dimensional function, it is obvious that we should use partial derivatives \(\partial/\partial x_1, \partial/\partial x_2, \ldots, \partial/\partial x_N\) rather than simple derivative \(\mathrm{d}/\mathrm{d}\xi\). To insert the partial derivatives, we need to use directional derivative. To proceed, consider a scalar function

in an \(\mathbb{E}^N\)-space, in which the coordinate axes are denoted by

Let

denote a vector pointing from a referencing point \(\mathbf{x}_r\) to \(\mathbf{x}\). The directional derivative of \(u\) is then written as

Similarly, higher-order directional derivatives can be written as

Aided by replacing the simple derivative in Eq. (1) with the directional derivative, the Taylor series of \(u\) about \(\mathbf{x}_r\) can be expressed as

Substituting Eq. (2) into Eq. (3) gives

Equation is the Taylor series of a multi-dimensional function \(u(x_1, x_2, \ldots, x_N)\).

It should be noted that symmetry of the higher-order partial derivatives is not assumed. That is, the mixed partial derivatives are not commutative.