Analogous notation to $\nabla$ but for gradient with respect to $\vec{k}$ not $\vec{x}$

Question

$\nabla = \frac{\partial}{\partial x_i}$ so $\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$.

However, is there a similar equalivalent notion for the vector $(\frac{\partial F}{\partial k_x}, \frac{\partial F}{\partial k_y}, \frac{\partial F}{\partial k_z})$ which could be useful, for instance, to express the group velocity which is given by $v_g = (\frac{\partial \omega}{\partial k_x},\frac{\partial \omega}{\partial k_y},\frac{\partial \omega}{\partial k_z})$. I know you could just use $\vec{v_g} = \frac{\partial \omega}{\partial k_i} \vec{x_i}$ but I was wondering about other notations.

Answer 1

If $\mathbf k=k_x\,\hat x+k_y\,\hat y+k_z\,\hat x$, then specifying the gradient of $F$ with respect to the variables $\{k_x,k_y,k_z\}$ can be denoted by

$$\nabla_{\mathbf k}F$$

Or possibly just

$$\nabla_kF$$

According to Wikipedia the transpose of the gradient can be expressed as $\text dF/\text d\mathbf k$.

Of course, I have seen these used to denote directional derivatives as well, so I think you just need to clarify your notation if using it and be careful about how it is being used when reading it.

I think I also remember Griffiths E&M using $\nabla'$ to denote the gradient with respect to primed coordinates, but I might be misremembering.