The field F = (x, y, z) has constant divergence ∇cdotF = 3. Use the divergence theorem to find its flux out of a cube of volume 8.

Practice question · Numerical answer

The field $\vec{F} = (x, y, z)$ has constant divergence $\nabla\cdot\vec{F} = 3$ . Use the divergence theorem to find its flux out of a cube of volume 8.

Hints

With constant divergence, the volume integral is just (divergence) × (volume).
$3 \times 8$ .

Show the answer

24 (answers within ±0.1 count)

Why

Flux $= 3 \times 8 = 24$ , no surface parametrization, no six cube faces, one multiplication. Converting hard boundary integrals into easy volume integrals (or vice versa) is exactly what the big theorems are for.

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The field F⃗=(x,y,z)\vec{F} = (x, y, z)F=(x,y,z) has constant divergence ∇⋅F⃗=3\nabla\cdot\vec{F} = 3∇⋅F=3. Use the divergence theorem to find its flux out of a cube of volume 8.

Practise The Divergence Theorem

More questions on The Divergence Theorem

The field $\vec{F} = (x, y, z)$ has constant divergence $\nabla\cdot\vec{F} = 3$ . Use the divergence theorem to find its flux out of a cube of volume 8.