Vector Fields: Divergence, Curl & Line Integrals

≈ 30 min

A vector field $\mathbf{F}(x, y) = (P, Q)$ attaches a vector to every point of the plane. Two scalar measurements describe its local behaviour.

Divergence (net outflow at a point):

\operatorname{div}\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}.

Curl (local rotation — in 2D a single scalar):

\operatorname{curl}\mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}.

A line integral $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r}$ measures the work done by $\mathbf{F}$ along a path $C$ . When $\mathbf{F} = \nabla\varphi$ for some potential $\varphi$ , the field is conservative and the integral is path-independent: $\int_C \mathbf{F}\cdot d\mathbf{r} = \varphi(\text{end}) - \varphi(\text{start}).$

At a point acting as a source, where flow spreads outward in all directions, the divergence is…

positive

negative

zero

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Find the divergence of $\mathbf{F}(x, y) = (x^2 y, \; xy^2)$ . Write your answer as an expression in $x$ and $y$ .

Find the (scalar) curl of the rotation field $\mathbf{F}(x, y) = (-y, \; x)$ .

The field $\mathbf{F} = (y, x)$ is conservative with potential $\varphi = xy$ . Find the work $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r}$ along any path from $(0, 0)$ to $(2, 3)$ .