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l_notes23

# l_notes23 - Lecture XXIII Surface Integrals Remember that...

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| Lecture XXIII Surface Integrals Remember that the parametric expression for a curve is given by a vector func- j + z ( t ) ˆ tion R ( t ) = x ( t ) ˆ i + y ( t ) ˆ k . The parametric expression for a surface S R ( u, v ) = x ( u, v ) ˆ i + y ( u, v ) ˆ is given by a vector function of two variables: j + z ( u, v ) ˆ 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 ˆ U around P ˆ to the surface area U around P . We can use the Jacobian to find the paramteric nromal vector: d R d R ( y, z ) ˆ ( x, z ) ˆ ( x, y ) i + j + ˆ 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 θ π , 0 z b , r = a . Clearly, since w P is normal to S , w P = r ˆ. Since Δ z and Δ θ in the plane correspond to Δ z | w P | and a Δ θ on the

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l_notes23 - Lecture XXIII Surface Integrals Remember that...

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