Gradient of the electric potential

Question

I was wondering if for a point-like charged object, does the gradient of the electric potential point in the direction of maximum increase or maximum decrease of the function $V$?

Answer 1

I'd like to add a bit of mathematical detail the (correct) statements by DJBunk. Let a scalar function $f$ be given (let's not restrict ourselves to the electric potential). For any unit vector $\mathbf n$, we can define the directional derivative $D_\mathbf{n}$ of the function $f$ in the direction $\mathbf n$ as follows: $$ D_\mathbf{n}f(\mathbf x) = \mathbf n\cdot\nabla f(x). $$ The directional derivative gives the rate of change of the scalar function $f$ in the direction of the unit vector $\mathbf n$. Notice that $$ \mathbf n\cdot \nabla f(\mathbf x) = |\nabla f(\mathbf x)|\cos\theta $$ where $\theta$ is the angle between $\mathbf n$ and $\nabla f(\mathbf x)$, so the directional derivative is maximized when $\theta = 0$, and is minimized when $\theta = -\pi$. In other words;

The the rate of change of a scalar function $f$ at a point $\mathbf x$ is positive and greatest in magnitude in the direction of the gradient of $f$ at $\mathbf x$.

This confirms BJBunk's statements.