MAST20029 · Engineering Mathematics
Gauss' Divergence Theorem
Gauss' (divergence) theorem converts a flux integral over a closed surface into a volume integral of the divergence: ∭_V ∇·F dV = ∬_S F·n̂ dS, with S the outward-oriented boundary of V. It is the exam's flux-over-a-closed-surface workhorse (a recurring Q3 / MST question): rather than integrating over several faces you compute one divergence and one volume integral, almost always in cylindrical or spherical coordinates.
What this chapter covers
- 011. The statement ∭_V ∇·F dV = ∬_S F·n̂ dS, S the closed boundary with outward normal
- 022. When it applies: a CLOSED surface (include all caps/faces) and outward orientation
- 033. The strategy: compute ∇·F, then a volume integral instead of face-by-face flux
- 044. Choosing coordinates for the volume integral (cylindrical / spherical) by symmetry
- 055. Carrying the Jacobian (r or r² sin φ) into the converted volume integral
- 066. Closing an open surface by adding the missing cap, or accounting for it
- 077. Sign of the normal: outward for the standard statement
- 088. Reading the trigger — 'flux out of a closed surface' means reach for Gauss
Flux out of a sphere by the divergence theorem
- 1 markRecognise a CLOSED surface with outward normal, so apply Gauss' theorem. Compute the divergence: ∇·F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 1 + 1 + 1 = 3.
- 2 marksConvert to a volume integral over the solid ball: flux = ∭_V 3 dV = 3 · Volume(ball of radius a) = 3 · (4/3)π a³ = 4π a³.
- 2 marksCheck by direct setup (optional) in spherical coordinates: ∭_V 3 dV = 3 ∫₀²π ∫₀^π ∫₀ᵃ r² sin φ dr dφ dθ = 3 · (a³/3) · 2 · 2π = 4π a³, with the Jacobian r² sin φ.
- 1 markState the flux: 4π a³.
Key terms
- Gauss' divergence theorem
- ∭_V ∇·F dV = ∬_S F·n̂ dS, equating the volume integral of the divergence to the outward flux through the closed surface S bounding V.
- Closed surface
- A surface with no boundary edge that fully encloses a solid (a sphere, or a cylinder WITH its top and bottom disks). Gauss' theorem requires the surface to be closed.
- Outward normal n̂
- The unit normal pointing out of the enclosed solid; it is the orientation assumed in the standard statement of the divergence theorem.
- Flux ∬_S F·n̂ dS
- The net flow of the vector field F through the surface S. Gauss' theorem lets you compute it as a single volume integral of ∇·F.
- Volume integral conversion
- Replacing a multi-face flux integral by ∭_V ∇·F dV — usually evaluated in cylindrical or spherical coordinates chosen to match the solid's symmetry.
- Closing an open surface
- Adding the missing cap (e.g. the lid of a hemisphere) so Gauss applies, then subtracting that cap's flux if it was not part of the original surface.
Gauss' Divergence Theorem FAQ
When do I use Gauss' theorem instead of integrating directly?
Whenever you need the flux through a closed surface and computing the divergence is easier than integrating over each face. The theorem turns a multi-face surface integral into one volume integral of ∇·F, which is usually far quicker — especially when the divergence is simple or constant.
What if the surface is not closed?
Gauss requires a closed surface, so either close it by adding the missing cap (then subtract that cap's flux at the end if it was not part of the original surface) or, if only the flux through a closed surface is asked, make sure you have included every face. A hemisphere question, for instance, usually needs its flat disk added.
Which coordinate system should the volume integral use?
Match it to the solid's symmetry: a cylinder calls for cylindrical coordinates (Jacobian r), a ball or sphere for spherical coordinates (Jacobian r² sin φ). Remember to carry the Jacobian into the converted integral — it rides along from the change of variables just as in the multiple-integral chapter.
How is Gauss' theorem examined?
It is a recurring full question (often Q3, and a mid-semester-test favourite): you are given a field and a closed surface and asked for the outward flux. State the theorem by name, compute ∇·F, convert to a volume integral in the right coordinates with the Jacobian, and evaluate. Forgetting to close an open surface or dropping the Jacobian are the usual mark-losers.
Exam move
Train the trigger: 'flux through a closed surface' should make you reach for Gauss before you set up any face integral. Write the theorem, compute ∇·F (note when it is constant, which collapses the volume integral to constant × volume), and convert to a volume integral in the coordinate system matched to the solid, carrying the Jacobian. Be vigilant about closure — an open surface must be closed by adding the missing cap, then corrected at the end. Practise on cylinders and spheres from past papers so that setting limits and choosing coordinates is automatic, and always name the theorem you are applying.
