Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro Multivariate Calculus - Bruce E. Shapiro
194 LECTURE 23. GREEN’S THEOREM Example 23.2 Let C be a circle of radius a centered at the origin. Find ∮ C F · dr for F = (−x 2 y, xy 2 ) using Green’s theorem. Solution. Since M = −x 2 y and N = xy 2 , we have M y = −x 2 , N x = y 2 From Green’s theorem ∮ F · dr = (N x − M y )dA = (x 2 + y 2 )dA Converting to polar coordinates ∮ F · dr = C C R ∫ 2π ∫ a 0 0 r 3 drdθ = ∫ 2π 0 R a 4 4 dθ = a4 π 2 Theorem 23.2 Gauss’ Theorem on a Plane Let C be a simple closed curve enclosing a region S ⊂ R 2 , and let F = (M(x, y), N(x, y)) be a smoothly differentiable vector field on S. Then ∮ F · nds = ∇ · FdA (23.9) where n(x, y) is a nunit tangent vector at (x, y). C Proof. Suppose that C is described by a parameter t as Recall that we derived earlier that if s is the arclength, T = dr ( dx ds = ds , dy ) (23.10) ds is a unit tangent vector, and an outward pointing unit normal vector is ( ) dy n = ds , −dx ds Hence ∮ C ∮ F · nds = ∮ = C C = R ∫ = R S ( ) dy (M(x, y), N(x, y)) · ds , −dx ds ds (M(x, y)dy − N(x, y)dx) ( ∂M ∂x + ∂N ) dA ∂y ∇ · FdA (23.11) Revised December 6, 2006. Math 250, Fall 2006
Lecture 24 Flux Integrals & Gauss’ Divergence Theorem A classic example of flux is the flow of water. Suppose water is flowing through a hole - say a window in a wall - of area A at some velocity v which makes an angle θ with the window’s normal vector. Then the volume of water that passes thorough the widow during a time interval dt is V = A‖v‖ cos θdt Figure 24.1: Flow of fluid through a window Definition 24.1 Let S be a surface, and dA an infinitesimal surface element. Then the area vector of S is dA = ndA where n is a unit normal vector to the surface element. A surface is said to be oriented if an area vector has been defined for its surface. Note that there are two possible orientations for any surface, because the surface unit normal can point in either direction. For a closed surface, these would correspond to output and inward pointing normal vectors. 195
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Lecture 24<br />
Flux Integrals & Gauss’<br />
Divergence Theorem<br />
A classic example of flux is the flow of water. Suppose water is flowing through a<br />
hole - say a window in a wall - of area A at some velocity v which makes an angle θ<br />
with the window’s normal vector. Then the volume of water that passes thorough<br />
the widow during a time interval dt is<br />
V = A‖v‖ cos θdt<br />
Figure 24.1: Flow of fluid through a window<br />
Definition 24.1 Let S be a surface, and dA an infinitesimal surface element. Then<br />
the area vector of S is<br />
dA = ndA<br />
where n is a unit normal vector to the surface element. A surface is said to be<br />
oriented if an area vector has been defined for its surface.<br />
Note that there are two possible orientations for any surface, because the surface<br />
unit normal can point in either direction. For a closed surface, these would<br />
correspond to output and inward pointing normal vectors.<br />
195