Definition:Taylor Series/Two Variables

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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:

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

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