clarification of notation for partial derivatives

Question

its been a while since i did vector calculus and i am a little bit rusty on the notation, can someone please tell me if this is true:

$\nabla_\mathbf{a} C = \dfrac{\partial C}{\partial \mathbf{a}}$

ie. the partial derivative of $C$ with respect to $\mathbf{a}$

or am i getting things mixed up?

Answer 1

Both the left and the right are commonly used notations for the directional derivative of $f$ in the direction of the vector $\mathbf a$. When $f$ is differentiable, this is, of course, given by dotting the gradient of $f$ with $\mathbf a$.