Recipe

Apply the Divergence Theorem to a closed-surface flux integral

Use when: you need \oiint_S\mathbf{F}\cdot d\mathbf{S} for a closed surface S fully enclosing a solid E

Goal: the flux value, computed as a triple integral of divergence over E instead of a surface integral over S

Taught in The divergence theorem

1. 1

   Check that S fully encloses a solid E with no boundary rim, and that its stated orientation is outward.

   Why: the Divergence Theorem needs a genuinely closed surface; an open surface with a rim needs Stokes' Theorem instead, not this one.

2. 2

   Write \mathbf{F}=(P,Q,R) and identify the solid region E that S encloses.

   Example

   Let \mathbf{F}=(x^2,\,y^2,\,z^2) and let S be the boundary of the box E=[0,1]\times[0,2]\times[0,3], outward-oriented.

   Why: the theorem trades the surface's flux for an integral over exactly this solid.

3. 3

   Differentiate each component with respect to its own matching variable, then add all three.

   Example

   P_x=2x, Q_y=2y, R_z=2z, so \operatorname{div}\mathbf{F} = 2x+2y+2z.

   Why: this is the volume-density of outward flux that the theorem integrates over E.

4. 4

   Integrate \operatorname{div}\mathbf{F} over E, choosing coordinates that match E's shape.

   • E is a ball or built from spheres — spherical coordinates usually simplify the integral.
   • E is a cylinder or has circular cross-sections — cylindrical coordinates usually simplify it.
   • E is a box or another axis-aligned solid — rectangular coordinates are usually simplest.

   Why: the theorem only supplies the integrand; picking coordinates that match E's shape is what makes the triple integral tractable.

5. 5

   Carry out the integration over E's actual bounds. This is the final flux value.

   Example

   \displaystyle\iiint_E (2x+2y+2z)\,dV = \int_0^1\!\!\int_0^2\!\!\int_0^3 (2x+2y+2z)\,dz\,dy\,dx = 36.

   Why: this number equals the original closed-surface flux integral exactly, with no surface parametrization ever required.