Solution
= Solution
The <total derivative> of $F$ at $x$ is the <linear map> $DF_x:\mathbb R^n\to\mathbb R^m$ such that
$$
\boxed{F(x+h)=F(x)+DF_x(h)+o(\|h\|).}
$$
The <Jacobian matrix> is the $m\times n$ <matrix> $J_F(x)=(\partial F_i/\partial x_j)(x)$. When the total <derivative> exists, this is its <matrix> in standard bases: $DF_x(h)=J_F(x)h$. Existence of partial <derivatives> alone need not imply existence of the total <derivative>.