20e-lecture24

# 20e-lecture24 - Lecture 24 Friday May 29th [email protected]

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Unformatted text preview: Lecture 24 - Friday May 29th [email protected] Key words: Surface integrals of vector fields 24.1 Surface integrals of vector fields We now come to surface integrals of vector fields. The motivation for line integrals of vector fields was the physical notion of work; here the motivation is flux. If a vector field f denotes the velocity vector field of fluid flowing through a surface, then the flux is the amount of fluid flowing through the surface per unit time. The flux will precisely be defined by the surface integral of the vector field. Definition. Let f : R 3 → R 3 be a vector field defined on a surface S parametrized by R ( u,v ) = ( x ( u,v ) ,y ( u,v ) ,z ( u,v )) for ( u,v ) ∈ D with tangent vectors T u and T v . Then the surface integral of f over S is defined by ZZ S f · dR = ZZ D f ( R ( u,v )) · ( T u × T v ) dudv when this integral exists. The flux through S for a fluid with velocity field f is RR S f · dS . Example 1. Compute the flux across the unit hemisphere if w ( x,y,z ) = (0 , ,z ) denotes the velocity vector field of fluid flowing through the hemisphere. Solution. We parametrize the hemisphere as R ( x,y ) = ( x,y, p 1- x 2- y 2 ) for x 2 + y 2 ≤ 1. Then T x = ‡ 1 , ,- x p 1- x 2- y 2 · T y = ‡ , 1 ,- y p 1...
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20e-lecture24 - Lecture 24 Friday May 29th [email protected]

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