# Definition:Taylor Series/Two Variables

## Definition

Let $f: \R^2 \to \R$ be a real-valued function of $2$ variables which is smooth on the open rectangle $\left({a \,.\,.\, b}\right) \times \left({c \,.\,.\, d}\right)$.

Let $\left({\xi, \zeta}\right) \in \left({a \,.\,.\, b}\right) \times \left({c \,.\,.\, d}\right)$.

Then the Taylor series expansion of $f$ about $\left({\xi, \zeta}\right)$ is:

 $\ds f \left({x, y}\right)$ $=$ $\ds f \left({\xi, \zeta}\right) + \left({x - \xi}\right) f_x \left({\xi, \zeta}\right) + \left({y - \zeta}\right) f_y \left({\xi, \zeta}\right)$ $\ds$  $\, \ds + \,$ $\ds \frac 1 {2!} \left({\left({x - \xi}\right)^2 f_{xx} \left({\xi, \zeta}\right) + 2 \left({x - \xi}\right) \left({y - \zeta}\right) f_{xy} \left({\xi, \zeta}\right) + \left({y - \zeta}\right)^2 f_{yy} \left({\xi, \zeta}\right) }\right)$ $\ds$  $\, \ds + \,$ $\ds \cdots$

where $f_x \left({\xi, \zeta}\right)$, $f_y \left({\xi, \zeta}\right)$ denote partial derivatives with respect to $x, y, \ldots$ evaluated at $x = \xi$, $y = \zeta$.