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Unformatted text preview: | Lecture XXIII Surface Integrals Remember that the parametric expression for a curve is given by a vector func- j + z ( t ) tion R ( t ) = x ( t ) i + y ( t ) k . The parametric expression for a surface S R ( u, v ) = x ( u, v ) i + y ( u, v ) is given by a vector function of two variables: j + z ( u, v ) k . The following are two fundamental properties of the parametric normal vector w ( u, v ): 1. If w P is normal to S at P . w P = 0, then 2. | w P | is an amplification factor from the area U around P to the surface area U around P . We can use the Jacobian to find the paramteric nromal vector: dR dR ( y, z ) ( x, z ) ( x, y ) i + j + k. w ( u, v ) = = P ( u, v ) ( u, v ) ( u, v ) du dv P Consider the semicylinder S of radius a , with 0 z b and y 0. In cylindrical coordinates, it is described by 0 , 0 z b , r = a . Clearly, since w P is normal to S , w P = r . Since z and in the...
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