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Taylor Expansion: Taylor Series In Several Variables


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Taylor Series In Several Variables

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

: & Tx_1,\dots,x_d\

\righta_1,\dots,a_d \ x_j - a_j \ x_j - a_jx_k - a_k \ x_j - a_jx_k - a_kx_l - a_l + \dots

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

fa,b &+x-a\, f_xa,b +y-b\, f_ya,b \

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

\,,

Example

Firstly, we compute all partial derivatives we need

The Taylor series is

which in this case becomes

\Big 0x-0^2 + 2x-0y-0 + -1y-0^2 \Big + \cdots \

Since is analytic in y < 1, we have + \cdots

for y < 1.


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