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== 5.4#h(0.6em)Stokes' Theorem: Two Caps, One Boundary


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Objectives]


      - State Stokes' Theorem and pair the boundary's direction with the cap's normal by the right-hand rule.
- Verify that circulation around a rim equals curl flux through a flat cap, by direct computation.
- Use surface-independence to evaluate the flux through a curved cap without integrating over it.
]


    The last section computed a circulation of #math.equation(block: false, alt: "4.5 π")[$4.5 π$] around a circle and called it a hint. Here is the theorem it hints at. #strong[Stokes' Theorem] says that the circulation of a field around a closed curve equals the flux of its #strong[curl] through #emph[any] surface that has the curve as boundary:


    #math.equation(block: true, alt: "∮ sub ∂ S 𝐅 times d 𝐫 equals ∬ sub S open parenthesis ∇ times 𝐅 close parenthesis times d 𝐒 ,")[$∮_(∂ S) 𝐅 · d 𝐫 = ∬_(S) ( ∇ × 𝐅 ) · d 𝐒 ,$]


    provided the curve's direction and the surface's normal are paired by the right-hand rule. The startling word is #emph[any]. Hang two utterly different surfaces on the same rim and the theorem demands they collect identical curl flux — the cap is disposable, and only the boundary owns the answer. The figure stages exactly that confrontation.


    
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      A dark circle of radius #math.equation(block: false, alt: "1.5")[$1.5$] in the #math.equation(block: false, alt: "x y")[$x y$]-plane — the boundary — spanned by two translucent caps that share it as their rim: a flat yellow-toned disk and a purple-orange paraboloid dome, #math.equation(block: false, alt: "z equals 2.25 minus r squared")[$z = 2.25 − r^(2)$]. Threading both, a field of arrows circulates counterclockwise around the #math.equation(block: false, alt: "z")[$z$]-axis: #math.equation(block: false, alt: "𝐅 equals open parenthesis minus y , x , 0 close parenthesis")[$𝐅 = ( − y , x , 0 )$], the non-conservative field from the previous section, its components independent of height.
    
    
=== Explore


    + Orbit slowly and confirm the two facts everything below rests on: both surfaces hang on the #emph[same] dark rim, and the field looks identical at every height — no #math.equation(block: false, alt: "z")[$z$] appears in its components.
+ Look straight down the #math.equation(block: false, alt: "z")[$z$]-axis. The arrows circulate counterclockwise around the boundary. Curl your right hand's fingers that way: your thumb points #math.equation(block: false, alt: "plus z")[$+ z$]. That pairing — counterclockwise rim, upward normals — fixes every sign below.
+ The dome clearly has more surface area than the disk, and both sit in the same swirling field. Predict, with one sentence of reasoning: does more curl flux pass through the dome?
+ Compute #math.equation(block: false, alt: "∇ times 𝐅")[$∇ × 𝐅$] (the example below checks you). It comes out the same #emph[constant vertical] vector everywhere — which makes the disk's flux a mental computation and sets up the trap in step 3.
+ Orbit to a low side view and look at the dome near its rim, where it rises steeply — nearly vertical. How much of a purely vertical curl vector passes through a nearly vertical patch of surface? So where on the dome is its flux actually collected?
+ Suppose the rim were traversed clockwise seen from above. What happens to the circulation's sign, and which way must both caps' normals point for the theorem to still balance?


    
=== The boundary owns the answer


    Why should the cap not matter? Two caps on one rim, taken together (one normal flipped), enclose a solid region — and curl fields send zero net flux out of any closed surface. So whatever flux enters through one cap leaves through the other: hang any surface on the rim and the total is fixed by the rim alone. That is why the theorem writes #math.equation(block: false, alt: "∂ S")[$∂ S$] on one side: only boundary data survives. In practice you read the equation right-to-left — replace a hard surface with an easy one, or with the boundary curve itself.


    #examplebox("Example 1")[One rim, one number, three ways][
For #math.equation(block: false, alt: "𝐅 equals open parenthesis minus y , x , 0 close parenthesis")[$𝐅 = ( − y , x , 0 )$] and the circle #math.equation(block: false, alt: "x squared plus y squared equals 2.25")[$x^(2) + y^(2) = 2.25$] in the #math.equation(block: false, alt: "x y")[$x y$]-plane, oriented counterclockwise from above: compute the curl, the flux through the flat disk, the circulation around the rim, and deduce the flux through the paraboloid dome #math.equation(block: false, alt: "z equals 2.25 minus r squared")[$z = 2.25 − r^(2)$].


      The curl of #math.equation(block: false, alt: "open parenthesis P , Q , R close parenthesis equals open parenthesis minus y , x , 0 close parenthesis")[$( P , Q , R ) = ( − y , x , 0 )$] is


        #math.equation(block: true, alt: "∇ times 𝐅 equals open parenthesis R sub y minus Q sub z , P sub z minus R sub x , Q sub x minus P sub y close parenthesis equals open parenthesis 0 , 0 , 1 minus open parenthesis minus 1 close parenthesis close parenthesis equals open parenthesis 0 , 0 , 2 close parenthesis ,")[$∇ × 𝐅 = ( R_(y) − Q_(z) , P_(z) − R_(x) , Q_(x) − P_(y) ) = ( 0 , 0 , 1 − ( − 1 ) ) = ( 0 , 0 , 2 ) ,$]


        the same vertical vector at every point in space. The disk's upward unit normal is #math.equation(block: false, alt: "𝐤")[$𝐤$], so its flux needs no integral: #math.equation(block: false, alt: "2 times area equals 2 π open parenthesis 1.5 close parenthesis squared equals 4.5 π")[$2 × "area" = 2 π ( 1.5 )^(2) = 4.5 π$]. The rim circulation was the previous section's computation: #math.equation(block: false, alt: "𝐅 open parenthesis 𝐫 open parenthesis t close parenthesis close parenthesis times 𝐫 prime open parenthesis t close parenthesis equals 2.25")[$𝐅 ( 𝐫 ( t ) ) · 𝐫^(′) ( t ) = 2.25$] on #math.equation(block: false, alt: "𝐫 open parenthesis t close parenthesis equals open parenthesis 1.5 cos t , 1.5 sin t , 0 close parenthesis")[$𝐫 ( t ) = ( 1.5 cos t , 1.5 sin t , 0 )$], giving #math.equation(block: false, alt: "2.25 times 2 π equals 4.5 π")[$2.25 · 2 π = 4.5 π$]. Equality of the two is Stokes' Theorem, verified by hand. And the dome? It hangs on the same rim with the same orientation, so its curl flux is #emph[forced] to be #math.equation(block: false, alt: "4.5 π")[$4.5 π$] — not more, despite the extra area. Geometry agrees with the bookkeeping: near the rim the dome is nearly vertical, and a vertical patch collects nothing from a vertical curl vector; the dome gathers its #math.equation(block: false, alt: "4.5 π")[$4.5 π$] near the top, where it flattens out — and its shadow on the floor is the very disk we already integrated.
]


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Circulation around the boundary, curl flux through any cap — one number, owned by the rim. When a surface integral looks hopeless, Stokes' Theorem is permission to swap the surface for a better one, or for its boundary; the answer was never the cap's to begin with.
]