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 $\openint a b \times \openint c d$.

Let $\tuple {\xi, \zeta} \in \openint a b \times \openint c d$.

Then the Taylor series expansion of $f$ about $\tuple {\xi, \zeta}$ is:

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