TITLE: What is the gradient with respect to a vector $\mathbf x$?
QUESTION [18 upvotes]: What is the meaning of "gradient with respect to $\mathbf x$"?
http://en.wikipedia.org/wiki/Gradient
I am talking about the symbol $$\nabla_\mathbf x$$
Does that simply mean derivative with respect to $\mathbf x$?

REPLY [11 votes]: Let $f=f(\boldsymbol x,\boldsymbol u)=f(x_1,...,x_n,u_1,...,u_r)$. Then 
$$\nabla f=\left(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n},\frac{\partial f}{\partial u_1},...,\frac{\partial f}{\partial u_r} \right),$$
$$\nabla _{\boldsymbol x}f=\left(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n} \right),$$
and 
$$\nabla _{\boldsymbol u}f=\left(\frac{\partial f}{\partial u_1},...,\frac{\partial f}{\partial u_r} \right).$$
so that $\nabla f(\boldsymbol x,\boldsymbol u) = (\nabla_{\mathbf{x}}f(\boldsymbol x,\boldsymbol u),\nabla_{\mathbf{u}}f(\boldsymbol x,\boldsymbol u))$