Lecture 25 Double integrals F dA Example x dy Mass Center of mass centroid x y Moment of inertia L a line p distance to L I Polar coordinates I Change of variables formula When L is the z axis we get I which is the polar moment of inertia x rcos0 and y rsin0 and dA rdrd0 Example R the unit circle about zero and density is one Find the polar moment of inertia When u u s t and v v s t and dudv det dsdt Example x rcos0 and y rsin0 and dxdy det drd0 det drd0 Line integrals with a curve C and Vector Field F M N Work Fundamental theorem of calculus for the gradient of vector elds Example F x y and C y x we need to parameterize C t t 0 t 1 W Green s Theorem curl M N N M curl Vf 0 f f f f 0 Converse F is de ned everywhere and curl F 0 then F Vf Example Let C be the unit circle which is cost sint when parameterized and let F x x Solving using Green s theorem and the double integral Flux form of Green s Theorem where Flux
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