For the vector valued function F = (x^2-y^2)i + 2xyj,
how do I find the matrix for the gradient of F?
If $\displaystyle {\bf F}(x,y)=f_1(x,y){\bf i}+f_2(x,y){\bf j}$, then
$\displaystyle \nabla {\bf F}=\left(\begin{array}{cc} \partial_x f_1 & \partial_y f_1 \\ \partial_x f_2 & \partial_y f_2 \end{array}\right)$
where
$\displaystyle \partial_x f=\frac{\partial f}{\partial x}$ etc.