Finals Notes

Calculus Andre Sostar 2014 Contents 1 Substitution in multiple integrals 3 Polar coordinates 3 Cylindrical coordinates 3 Spherical coordinates 4 2 Vector elds and line integrals 5 Dot and Cross product of a vector eld 5 Gradient of a function 5 Directional derivative, Divergence, Curl 5 3 Line integral 6 Line integral 6 Work of a vector eld 6 4 Newton-Leibniz' and Green's Theorems 7 The Newton-Leibniz' Theorem 7 Conservative vector elds 7 Green's Theorem 8 1 5 Parametrization 9 Parametrization 9 2 1 Substitution in multiple integrals Polar coodrinates: r, th are dened as: x = r cos th, y = r sin th r2 = x2 + y 2 , r = r cos2 th + r sin2 th dA = r * drdth Finally where [?][?] f (x, y)dxdy = D [?][?] r * f (r cos th, r sin th) drdth E E is the same region in r, th. Cylindrical coordinates: r, th, z are dened as: x = r cos th, y = r sin th, z = z r2 = r2 cos2 th + r2 sin2 th dV = rdr dthdz Finally where [?][?][?] f (x, y, z)dxdydz = V [?][?][?] r * f (r cos th, r sin th, y) drdthdz W W is the same region in cylindrical coordinates. 3

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