MAST20029 · Engineering Mathematics
Stokes' Theorem
Stokes' theorem relates the flux of the curl over an open surface to the circulation of the field around its boundary: ∬_S (∇×F)·n̂ dS = ∮_C F·dr, with the surface normal and the boundary orientation tied together by the right-hand rule. It lets you swap a hard surface integral for an easier boundary line integral (or vice versa), and it makes the conservative-field result (zero curl ⇒ zero circulation) transparent.
What this chapter covers
- 011. The statement ∬_S (∇×F)·n̂ dS = ∮_C F·dr (open surface S, boundary curve C)
- 022. Open surface vs closed: Stokes needs a boundary curve (Gauss needs closure)
- 033. The right-hand rule linking the normal n̂ to the orientation of C
- 044. Choosing the easier side: compute the curl flux OR the boundary circulation
- 055. Replacing a complicated cap by any surface with the same boundary
- 066. Parametrising the boundary curve C to evaluate ∮_C F·dr
- 077. Link to conservative fields: curl F = 0 ⇒ circulation = 0 round any loop
- 088. Keeping orientation consistent so the two sides have matching signs
Circulation via Stokes' theorem
- 3 marksCompute the curl: ∇×F has i-component ∂(z)/∂y − ∂(x)/∂z = 0, j-component −(∂(z)/∂x − ∂(−y)/∂z) = 0, and k-component ∂(x)/∂x − ∂(−y)/∂y = 1 − (−1) = 2. So ∇×F = 2 k.
- 2 marksChoose the surface S = the flat disk with upward normal n̂ = k, consistent with the anticlockwise C by the right-hand rule. Then (∇×F)·n̂ = (2 k)·k = 2.
- 1 markApply Stokes: ∮_C F·dr = ∬_S (∇×F)·n̂ dS = ∬_S 2 dS = 2 · area(disk) = 2 · π(1)² = 2π.
- 1 markState the circulation: 2π.
Key terms
- Stokes' theorem
- ∬_S (∇×F)·n̂ dS = ∮_C F·dr: the flux of the curl through an open surface equals the circulation of F around its boundary curve C.
- Open surface
- A surface with a boundary edge (a disk, a hemisphere cap). Stokes' theorem applies to open surfaces; the boundary curve C is what the line integral runs over.
- Boundary curve C
- The edge ∂S of an open surface S. Its orientation must be compatible with the surface normal n̂ by the right-hand rule for Stokes' theorem to hold with the correct sign.
- Right-hand rule
- The orientation convention: curl the right hand's fingers in the direction of C and the thumb points along n̂. It ties the boundary direction to the surface normal so both sides of Stokes agree in sign.
- Circulation ∮_C F·dr
- The line integral of F around the closed boundary curve C, measuring net flow along the loop. Stokes equates it to the flux of the curl through any surface bounded by C.
- Surface independence
- Stokes lets you replace the given cap by any other surface with the same boundary curve, since both give the same curl flux — choose the simplest (often the flat disk).
Stokes' Theorem FAQ
What is the difference between Stokes' theorem and Gauss' theorem?
Gauss relates a CLOSED surface's flux to a volume integral of the divergence; Stokes relates an OPEN surface's curl flux to a line integral around its boundary. Gauss needs no boundary curve (closed surface); Stokes needs one (open surface). Read whether the surface is closed to choose the right theorem.
How do I orient the boundary curve correctly?
Use the right-hand rule: if the normal n̂ points up out of the surface, the boundary C is traversed anticlockwise as seen from the normal side. Get this wrong and your two sides differ by a sign. Stating the chosen normal and the matching orientation earns marks and prevents sign errors.
Can I change the surface in a Stokes calculation?
Yes — that is one of its main uses. The curl flux is the same through any surface sharing the same boundary curve C, so replace a complicated cap with the simplest surface (often the flat disk) to make (∇×F)·n̂ easy. Just keep the boundary and orientation fixed.
How does Stokes relate to conservative fields?
If F is conservative then curl F = 0, so the left side of Stokes is zero and therefore the circulation ∮_C F·dr around any closed loop is zero. This is exactly the path-independence property from the conservative-fields chapter, viewed through Stokes' theorem.
Exam move
Decide first which side is easier: if the curl is simple, compute the curl flux over the surface; if the boundary is a simple curve, compute the circulation directly. Always state the surface normal and confirm the boundary orientation by the right-hand rule before evaluating, since a wrong orientation flips the sign. Exploit surface independence — replace an awkward cap by the flat disk with the same boundary — and remember that a conservative field gives zero circulation. Practise the curl computation (with its + − + sign pattern) and a clean boundary parametrisation, and name the theorem you use.
