散度定理

The third fundamental theorem

A worked cube

The radial field, revisited with power

Capping an open surface

A student replaces the flux integral over an open bowl by a triple integral of divergence over the solid beneath it. What is missing?

$\mathbf{F} = (x, y, z)$ and a solid has volume $10$. What is the flux out of its boundary, whatever the shape?

A field has divergence $-2$ throughout a region. What does flux out of the region's boundary report?

$\mathbf{F} = (x^2, y^2, z^2)$ over the unit cube. Give the total outward flux, as a plain number.

The paraboloid $z = x^2 + y^2$ below $z = 1$, with $\mathbf{F} = (x, y, z)$, is capped by the disk at $z = 1$. The closed flux is $3\pi/2$ and the lid's flux is $\pi$. Give the bowl's outward flux, in the form a*pi/b.