information:filtering
Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with

T (x 1, …, x d) = ∑ n 1 = 0 ∞ ⋯ ∑ n d = 0 ∞ (x 1 − a 1) n 1 ⋯ (x d − a d) n d n 1 ! ⋯ n d ! (∂ n 1 + ⋯ + n d f ∂ x 1 n 1 ⋯ ∂ x d n d) (a 1, …, a d) = f (a 1, …, a d) + ∑ j = 1 d ∂ f (a 1, …, a d) ∂ x j (x j − a j) + 1 2 ! ∑ j = 1 d ∑ k = 1 d ∂ 2 f (a 1, …, a d) ∂ x j ∂ x k (x j − a j) (x k − a k) + 1 3 ! ∑ j = 1 d ∑ k = 1 d ∑ l = 1 d ∂ 3 f (a 1, …, a d) ∂ x j ∂ x k ∂ x l (x j − a j) (x k − a k) (x l − a l) + ⋯ {\displaystyle {\begin{aligned}T(x_{1},\ldots,x_{d})&=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots,a_{d})\\&=f(a_{1},\ldots,a_{d})+\sum _{j=1}^{d}{\frac {\partial f(a_{1},\ldots,a_{d})}{\partial x_{j}}}(x_{j}-a_{j})+{\frac {1}{2!}}\sum _{j=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}f(a_{1},\ldots,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})\\&\qquad \qquad +{\frac {1}{3!}}\sum _{j=1}^{d}\sum _{k=1}^{d}\sum _{l=1}^{d}{\frac {\partial ^{3}f(a_{1},\ldots,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\cdots \end{aligned}}}

For example, for a function f (x, y) {\displaystyle f(x,y)} that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is

f (a, b) + (x − a) f x (a, b) + (y − b) f y (a, b) + 1 2 ! ((x − a) 2 f x x (a, b) + 2 (x − a) (y − b) f x y (a, b) + (y − b) 2 f y y (a, b)) {\displaystyle f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)+{\frac {1}{2!}}{\Big (}(x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b){\Big)}}

where the subscripts denote the respective partial derivatives.