向量场与流线

A vector at every point

Gradient fields come from potentials

Flow lines follow the field

Fields versus their flow lines

The field $\mathbf{F}(x, y) = (-y, x)$ is sampled at the point $(1, 0)$. Which vector is attached there?

Which field is a gradient field, that is, of the form $\nabla f$ for some potential function $f$?

A particle follows the flow lines of $\mathbf{F}(x, y) = (-y, x)$ starting from $(1, 0)$. What path does it trace?

For the field $\mathbf{F}(x, y) = (2x, 2y)$, a potential $f$ satisfies $f_x = 2x$ and $f_y = 2y$. Give the value of $f(1, 1)$ for the potential $f = x^2 + y^2$, as a plain number.

The radial field $(x, y)$ has flow lines solving $x' = x$ and $y' = y$. A particle starts at $(2, 1)$. Give the ratio $y/x$ it maintains forever, as a plain number or fraction.