08 - Taylor's Theorem for a function of two variables

Taylor's Theorem for a function of two variables

Let $f(x,y)$ be defined on an open subset $U$, such that

• $f$ has continuous partial derivatives on $U$ up to the $(n+1)$th order

Let $(a,b)\in U$ and suppose the line segment between $(a,b)$ and $(x,y)$ lies in $U$

Then there exists $\theta \in (0,1)$ such that

$$\begin{array}{rl}f(x,y)& =f(a,b)+\sum _{k=1}^n\sum _{i+j=k,i,j\ge 0}\frac{1}{i!j!}\frac{{\partial }^{k}f(a,b)}{\partial x^i\partial y^j}(x-a{)}^i(y-b{)}^j\\ & +\sum _{i+j=n+1i,j\ge 0}\frac{1}{i!j!}\frac{{\partial }^{n+1}f(\xi )}{\partial x^i\partial y^j}(x-a{)}^i(y-b{)}^j\end{array}$$

where $\xi =\theta (a,b)+(1-\theta )(x,y)$

Note that $\xi$ is just $\theta$ of the line segment