1920 Final Review

842 WordsJul 26, 20144 Pages
Fundamental Theorem of Conservative Vector Fields: -The circulation around a closed path c is zero: -Cross Partials equal to zero and is simple connected -Has a potential function If a function meets any of these criteria it is simply connected. If not it is not. You can only really find the last two. Second one more so. Pg. 976 Very Important -Conservative Vector Fields have zero curl simple closed curve—that is, a closed curve that does not intersect itself Green’s Theorem Let D be a domain whose boundary ∂D is a simple closed curve, oriented counterclockwise. Then The surface in (C) is called a closed surface because its boundary is empty. To integrate cos^2θ use identity cos2 θ = 1/2 (1 + cos 2θ). Notice that curl(F) contains the partial derivatives ∂F1 and ∂F1 but not the partial ∂y ∂z ∂F1.SoifF1=F1(x)isafunctionofxalone,then∂F1 =∂F1 =0,andF1doesnot ∂x ∂y ∂z contribute to the curl. The same holds for the other components. In summary, if each of F1, F2, and F3 depends only on its corresponding variable x, y, or z, then curlF1(x), F2(y), F3(z) = 0 In other words, the surface integral of a vector field with vector potential A is surface independent, just as a vector field with a potential function V is path independent. 18.2 Stoke’s Theorem -The boundary orientation is the direction for which the surface is on your left as you walk. -The boundary of a surface is denoted ∂S -The boundary orientations in (B) are reversed because the opposite normal has been selected to orient the surface. Opposite direction of normal negates boundary orientation. - curl(F) = ∇ × F - The curl measures the extent to which F fails to be conservative. If F is conservative, then curl(F) = 0 and Stokes’ Theorem merely confirms what we already know: The circulation of a conservative vector field around a closed path is zero. - The curl of a vector field is

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