MTH2010 Chap.12 Stokes' Theorem and the Divergence Theorem
Stokes' Theorem and the Divergence Theorem
The final week of the unit lists Stokes' theorem, the divergence theorem and applications. Both theorems generalise Green's theorem from the previous chapter, and together with it they appear in the unit's learning outcomes as the three theorems named after Green, Stokes and Gauss. The curl of a vector field F = (P, Q, R) has components R_y - Q_z, P_z - R_x and Q_x - P_y, and its divergence is P_x + Q_y + R_z.
Stokes' theorem says that the circulation of F around the boundary curve of an oriented surface equals the flux of the curl of F through the surface. The orientations must match by the right-hand rule: with the thumb along the chosen normal, the fingers curl in the direction of travel around the boundary.
Because the right-hand side depends only on the boundary, any surface with the same boundary gives the same answer, which lets an awkward surface be replaced by a flat disc. The divergence theorem says that the outward flux of F through a closed surface equals the integral of the divergence over the solid it encloses. It turns six face integrals over a cube, or a curved integral over a sphere, into a single volume integral.
When a surface is not closed, adding a simple lid, applying the theorem and subtracting the lid's flux is a standard technique. Choosing between Green, Stokes and the divergence theorem depends on whether the object is a plane curve, a curve in space, or a closed surface.
What this chapter covers
- 01
Curl and divergence of a vector field
- 02
Stokes' theorem: circulation equals flux of the curl
- 03
Right-hand rule for matching orientations
- 04
Replacing a surface by a simpler one with the same boundary
- 05
Divergence theorem: outward flux equals the volume integral of divergence
- 06
Closing an open surface with a lid
- 07
Applications: flux of radial fields through spheres and cubes
- 08
Choosing between Green, Stokes and Gauss
Worked example · free
Outward flux through the unit sphere
- 1Divergence: 3x² + 3y² + 3z² = 3 ρ² in spherical coordinates.
- 1Volume integral: 3 ρ² times ρ² sin(φ), giving 3 times (1/5) times the angular factor 4 π.
- 1Flux = 12 π/5.
Key terms
- Curl
- The vector field built from the cross-derivatives of a field, measuring its local rotation.
- Stokes' theorem
- The result equating the circulation around a boundary curve with the flux of the curl through any surface it bounds.
- Divergence theorem
- The result equating outward flux through a closed surface with the volume integral of divergence inside it.
Stokes' Theorem and the Divergence Theorem FAQ
How do I match orientations in Stokes' theorem?
Use the right-hand rule. Point your thumb along the chosen normal of the surface; your fingers then curl in the direction the boundary must be traversed. Reversing either choice reverses the sign.
Can I change the surface in Stokes' theorem?
Yes, provided the new surface has the same boundary curve and a compatible orientation. The flux of the curl is the same for all such surfaces, so a flat disc is often the easiest choice.
What if the surface in a flux question is not closed?
The divergence theorem needs a closed surface. Close it with a simple surface such as a flat disc, apply the theorem to the whole closed surface, then subtract the flux through the added piece.
How do I decide which integral theorem to use?
A closed curve in the plane points to Green's theorem, a closed curve in space to Stokes' theorem, and a closed surface around a solid to the divergence theorem. An endpoint-only answer for a gradient field uses the fundamental theorem for line integrals.
Exam move
Memorise the curl and divergence formulas as patterns and compute them for a few standard fields until they are automatic. For Stokes, always draw the surface, its normal and the boundary arrow before calculating, and check one example both ways, as a circulation and as a flux of the curl. For the divergence theorem, practise on a cube and on a sphere, and rehearse the lid technique on a hemisphere.
Finish revision with a mixed set of questions where the first task is only to name the right theorem.
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