http://en.wikipedia.org/wiki/Taylor_series

Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with

For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:

where the subscripts denote the respective partial derivatives.

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

where  is the gradient of  evaluated at  and  is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes

which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case.

[edit]Example

Second-order Taylor series approximation (in gray) of a function  around origin.

Compute a second-order Taylor series expansion around point  of a function

Firstly, we compute all partial derivatives we need

The Taylor series is

which in this case becomes

Since log(1 + y) is analytic in |y| < 1, we have

for |y| < 1.