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,\backslash dots,x\_d\backslash $
\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\; \&+xa\backslash ,\; f\_xa,b\; +yb\backslash ,\; f\_ya,b\; \backslash $
A secondorder Taylor series expansion of a scalarvalued 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 0x0^2 + 2x0y0 + 1y0^2 \Big + \cdots \
Since is analytic in y < 1, we have + \cdots
for y < 1.
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