Taylor's Theorem/Two Variables

Theorem

Let $f: \R^2 \to \R$ be a real-valued function which is appropriately differentiable in a neighborhood of the point $\tuple {a, b}$.

Then for $\sqrt {h^2 + k^2} < R$ for some $R \in \R$:

\(\ds \map f {a + h, b + k}\)

\(=\)

\(\ds \frac 1 {0!} \map f {a, b}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \frac 1 {1!} \valueat {\paren {h \dfrac {\partial f} {\partial x} + k \dfrac {\partial f} {\partial y} } } {a, b}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \frac 1 {2!} \valueat {\paren {h^2 \dfrac {\partial^2 f} {\partial x^2} + 2 h k \dfrac {\partial^2 f} {\partial x \partial y} + k^2 \dfrac {\partial^2 f} {\partial y^2} } } {a, b}\)

\(\ds \)

\(\)

\(\, \ds + \, \)

\(\ds \cdots\)

Proof

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Source of Name

This entry was named for Brook Taylor.

Sources

• 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.4$ Differential calculus: $\text {(iv)}$ Taylor's series