Recipe
Apply Stokes' Theorem to a circulation integral
Use when: you need \oint_C\mathbf{F}\cdot d\mathbf{r} around a closed curve C in space, and a surface S with boundary C is given
Goal: the circulation value, computed as a curl-flux integral over S instead of a line integral around C
Taught in Stokes' theorem · a later section extends this recipe with more cases
Quick recipe
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1
Curl the fingers of your right hand in C's direction of travel; your thumb points the direction S's normal must be chosen.
Why: reversing either C's direction or S's chosen normal alone flips the sign of the identity; only a matched pair keeps it correct.
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2
Compute all three components of the curl.
Example
Let \mathbf{F}=(y,\,-x,\,z). Then \operatorname{curl}\mathbf{F} = (R_y-Q_z,\ P_z-R_x,\ Q_x-P_y) = (0,\,0,\,-1-1) = (0,0,-2).
Why: curl-flux through S is exactly the quantity Stokes' Theorem trades circulation for.
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3
Parametrize S, compute its cross product, and confirm (or flip) its direction to match Step 1.
Example
Let S be the flat disk of radius 2 in the plane z=0, matching C: the circle x^2+y^2=4, counterclockwise. Using \mathbf{r}(\rho,\theta)=(\rho\cos\theta,\rho\sin\theta,0), \mathbf{r}_\rho\times\mathbf{r}_\theta = (0,0,\rho), which already points in the +z direction matching Step 1's right-hand rule.
Why: this single computation supplies both the correctly directed normal and the area-stretch factor curl-flux needs, exactly as in the surface-integral chapter.
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4
Evaluate \operatorname{curl}\mathbf{F} on S, then dot it against the Step 3 cross product.
Example
\operatorname{curl}\mathbf{F}\cdot(0,0,\rho) = (0,0,-2)\cdot(0,0,\rho) = -2\rho.
Why: only the component of curl along the surface's normal contributes to curl-flux.
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5
Integrate the Step 4 result over S's full parameter domain. This is the final circulation value.
Example
\displaystyle\oint_C\mathbf{F}\cdot d\mathbf{r} = \int_0^{2\pi}\!\!\int_0^2 (-2\rho)\,d\rho\,d\theta = -8\pi.
- Not yet: choosing a DIFFERENT, easier surface sharing the same boundary when the given one is awkward to parametrize — covered in Using Stokes' theorem to change the surface
Why: this single number equals both the curl-flux over S and the original circulation integral around C, by Stokes' Theorem itself.
Common mistakes
- Choosing S's normal without checking it against C's direction via the right-hand rule.
- Computing curl with a sign error in one of the three cross-partial pairs.
- Trying to apply the theorem when C is not actually closed.
- Assuming S must be flat; any surface sharing the same boundary and orientation works.