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:

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