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4.5 Taylor’s theorem in two variables

Let $f(x,y)\,$ be a suitably differentiable function of two variables. Then $f$ has a multiple Taylor series involving the partial derivative of $f$ . The following theorem helps us to describe the behaviour of $f$ near to some point $P$ and is especially useful at stationary points.

Theorem.

Let $P=(a,b)\,$ be a point in the plane. Then for $h\,$ and $k\,$ sufficiently small

f(a+h,b+k)=f(a,b)+\Bigl\{h{{\partial f}\over{\partial x}}+k{{\partial f}\over{% \partial y}}\Bigr\}+\hskip{71.13189pt}

\hskip{28.452756pt}{{1}\over{2}}\Bigl(h^{2}{{\partial^{2}f}\over{{\partial x}^% {2}}}+2hk{{\partial^{2}f}\over{{\partial x}{\partial y}}}+k^{2}{{\partial^{2}f% }\over{{\partial y}^{2}}}\Bigr)+{\hbox{(higher order terms)}},

where the partial derivatives are evaluated at $P\,$ .