Let us say we have some differentiable vector field $F:\mathbb{R}^3 \to \mathbb{R}^3$. I have often seen the notation:
$$
\frac{\partial F}{\partial x} $$
Is this accepted notation? If so, is it usually a standin for the vector:
$$\left(\begin{matrix} \frac{\partial F_1}{\partial x} \\ \frac{\partial F_2}{\partial x}\\ \frac{\partial F_3}{\partial x} \end{matrix}\right)$$
Moreover, could we use the limit definition of the partial derivative to compute this derivative?
Places I have seen this notation are for example when one looks to calculate the surface area of a parametric $2$ – dim surface embedded in $\mathbb{R}^3$, where one takes the cross product of the partial derivative "vectors" of the map $u:\mathbb{R}^2 \to \mathbb{R}^3$ which parameterizes the surface.
Best Answer
If ${\bf f}(x)=\bigl(f_1(x),f_2(x),\ldots,f_m(x)\bigr)$ is a vector valued function of some variable $x$ then $$\lim_{x\to\xi}{\bf f}(x)={\bf a}\quad \Leftrightarrow\quad \lim_{x\to\xi} f_i(x)=a_i\quad(1\leq i\leq m)\ .$$ This is not a definition, but is true because of the way the metrics and convergence on ${\mathbb R}$ and ${\mathbb R}^m$ are defined. Rock bottom are the inequalities $$|y_i-a_i|\leq|{\bf y}-{\bf a}|\leq \sum_{i=1}^m |y_i-a_i|\ .$$ As a consequence we have $${\partial {\bf f}\over\partial x_k}({\bf x})=\left({\partial f_1\over\partial x_k},{\partial f_2\over\partial x_k},\ldots, {\partial f_m\over\partial x_k}\right)_{\bf x}\ ,$$ where on the LHS the vectorial limit $$\lim_{h\to0}{{\bf f}({\bf x}+ h{\bf e}_k)-{\bf f}({\bf x})\over h}$$ is meant.