18 02A Topic 48 Stokes theorem Author Jeremy Orloff Read SN V13 Analogy to Green s Theorem H RR Normal form of Green s Theorem F n ds R divF dA C RR RRR divergence theorem S F n dS D divF dV Tangential form of Green s Theorem Stokes Theorem Ingredients Examples H C F dr RR R curlF dA Let S be a surface with boundary curve C Note Since S is not closed and C is in space neither has a natural orientation We choose compatible 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 if you curl the fingers of your right hand around the curve then your thumb points in the direction of the normal Examples Stokes Theorem F a vector field S an oriented surface with boundary C compatibly oriented If F is continuously differentiable on S then I ZZ ZZ F dr curlF n dS F n dS C S S 1 18 02A topic 48 2 Example 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 2 Z 2 Z 2 I I 1 cos 2t cos t cos t dt 2z dx x dy y dz F dr dt 2 0 0 C C I We could also compute x dy using Green s Theorem since C is in the xyplane C Surface integral curlF h1 2 1i On S we have 1 n hx y zi x sin cos y sin sin z cos from 0 to 2 from 0 to 2 dV sin d d ZZ ZZ x 2y z dS curlF n dS S Z S Z 2 2 sin cos 2 sin sin cos sin d d Z0 2 Z0 2 sin2 cos 2 sin2 sin cos sin d d 0 0 We can see easily that the outer integral in the first two terms will be 0 So the integral Z 2 Z 2 becomes cos sin d d 0 0 2 Inner integral Outer integral sin2 2 0 1 2 2 1 2 Same as the line integral QED Example Let F x2 i xj z 2 k and let S be the graph of z x3 xy 2 y 4 above I the unit disk Use Stokes Theorem to compute F dr where C is the boundary of S C answer curlF h0 0 1i n dS h zx zy 1i dx dy curlF n dS dx dy ZZ ZZ curlF n dS dx dy area R S R I F dr Therefore by Stokes Theorem C Relation to Green s Theorem Green s Theorem is just a special case of Stokes where C 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 formula n 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 and make sure you understand it End of topic 48 notes
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