18.02A Topic 48: Stokes’ theorem.Author: Jeremy OrloffRead: SN: V13Analogy to Green’s TheoremNormal form of Green’s Theorem:HCF · n ds =RRRdivF dA divergence theorem:RRSF · n dS =RRRDdivF dV .Tangential form of Green’s Theorem:HCF · dr =RRRcurlF dA Stokes’ Theorem.Ingredients: Let S be a surface with boundary curve C.Examples:Note: Since S is not closed and C is in space neither has a natural orientation. We choosecompatible orientations by the right-hand rule.Right-hand rule: If you walk around C in the positive direction keeping S on your left,then your head points in the direction of n. This is called the right-hand rule because ifyou curl the fingers of your right hand around the curve then your thumb points in thedirection of the normal.Examples:Stokes’ TheoremF a vector field.S an oriented surface with boundary C, compatibly oriented.If F is continuously differentiable on S thenICF · dr =ZZScurlF · n dS =ZZS(∇ × F) · n dS.118.02A topic 48 2Example 1: Let S be the top half of the unit sphere and F = h2z, x, yi. Verify Stokes’Theorem.Line integral: Parametrize C: x = cos t, y = sin t, z = 0, t from 0 to 2π.⇒ICF · dr =IC2z dx + x dy + y dz =Z2π0cos t(cos t dt) =Z2π01 + cos 2t2dt = π.(We could also computeICx dy using Green’s Theorem since C is in the xyplane.)Surface integral: curlF = h1, 2, 1i.On S we have ρ = 1 ⇒ n = hx, y, zi, x = sin φ cos θ, y = sin φ sin θ, z = cos φ,φ from 0 to π/2, θ from 0 to 2π, dV = sin φ dφ dθ.⇒ZZScurlF · n dS =ZZSx + 2y + z dS=Z2π0Zπ/20(sin φ cos θ + 2 sin φ sin θ + cos φ) sin φ dφ dθ=Z2π0Zπ/20sin2φ cos θ + 2 sin2φ sin θ + cos φ sin φ dφ dθ.We can see easily that the outer integral in the first two terms will be 0. So, the integralbecomesZ2π0Zπ/20cos φ sin φ dφ dθ.Inner integral:sin2φ2π/20=12.Outer integral:12· 2π = π. Same as the line integral, QED.Example: Let F = x2i + xj + z2k and let S be the graph of z = x3+ xy2+ y4abovethe unit disk. Use Stokes’ Theorem to computeICF · dr, where C is the boundary of S.answer: curlF = h0, 0, 1i, n dS = h−zx, −zy, 1i dx dy ⇒ curlF · n dS = dx dy.⇒ZZScurlF · n dS =ZZRdx dy = area R = π.Therefore, by Stokes’ TheoremICF · dr = π.Relation to Green’s Theorem: Green’s Theorem is just a special case of Stokes’ whereC is a curve in the plane with interior R, and n = k, curlF = (Nx− My)k.Proof of Stokes’ Theorem: The proof is a careful use of the chain rule and the formulan dS = h−fx, −fy, 1i dx dy to reduce Stokes’ formula to Green’s.If you really want to master the subject you should read the proof in the notes (V13) andmake sure you understand it.End of topic 48
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