Taylor's theorem in two variables

Adding the three second-order terms ( $f_{xx}$ , $f_{xy}$ , $f_{yy}$ ) to the tangent plane produces a paraboloid that also matches the surface's curvature.

The second-order Taylor polynomial $P_2(x,y)$ is the tangent plane $L(x,y)$ plus three quadratic correction terms built from the three distinct second partials $f_{xx}$ , $f_{xy}=f_{yx}$ (Clairaut's theorem is why there are three terms here and not four), and $f_{yy}$ : $P_2(x,y)=L(x,y)+\tfrac12 f_{xx}(a,b)(x-a)^2+f_{xy}(a,b)(x-a)(y-b)+\tfrac12 f_{yy}(a,b)(y-b)^2$ . Where $L(x,y)$ only guarantees the approximation error shrinks faster than distance, $P_2(x,y)$ guarantees it shrinks faster than distance squared.

Examples

For $f(x,y)=x^2+xy+y^2$ at $(1,1)$ , $P_2(x,y)=3+3(x-1)+3(y-1)+(x-1)^2+(x-1)(y-1)+(y-1)^2$ , and since $f$ is already degree 2, $P_2$ equals $f$ exactly.