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Unformatted text preview: 68 Theory Supplement Section M M PROOF OF THE DIVERGENCE THEOREM AND STOKES’ THEOREM In this section we give proofs of the Divergence Theorem and Stokes’ Theorem using the definitions in Cartesian coordinates. Proof of the Divergence Theorem Let ~ F be a smooth vector field defined on a solid region V with boundary surface A oriented outward. We wish to show that Z A ~ F · d ~ A = Z V div ~ F dV. For the Divergence Theorem, we use the same approach as we used for Green’s Theorem; first prove the theorem for rectangular regions, then use the change of variables formula to prove it for regions parameterized by rectangular regions, and finally paste such regions together to form general regions. Proof for Rectangular Solids with Sides Parallel to the Axes Consider a smooth vector field ~ F defined on the rectangular solid V : a ≤ x ≤ b , c ≤ y ≤ d , e ≤ z ≤ f . (See Figure M.50). We start by computing the flux of ~ F through the two faces of V perpendicular to the x-axis, A 1 and A 2 , both oriented outward: Z A 1 ~ F · d ~ A + Z A 2 ~ F · d ~ A =- Z f e Z d c F 1 ( a, y, z ) dy dz + Z f e Z d c F 1 ( b, y, z ) dy dz = Z f e Z d c ( F 1 ( b, y, z )- F 1 ( a, y, z )) dy dz. By the Fundamental Theorem of Calculus, F 1 ( b, y, z )- F 1 ( a, y, z ) = Z b a ∂F 1 ∂x dx, so Z A 1 ~ F · d ~ A + Z A 2 ~ F · d ~ A = Z f e Z d c Z b a ∂F 1 ∂x dx dy dz = Z V ∂F 1 ∂x dV. By a similar argument, we can show Z A 3 ~ F · d ~ A + Z A 4 ~ F · d ~ A = Z V ∂F 2 ∂y dV and Z A 5 ~ F · d ~ A + Z A 6 ~ F · d ~ A = Z V ∂F 3 ∂z dV. Adding these, we get Z A ~ F · d ~ A = Z V µ ∂F 1 ∂x + ∂F 2 ∂y + ∂F 3 ∂z ¶ dV = Z V div ~ F dV. This is the Divergence Theorem for the region V . Theory Supplement Section M 69 x y z A 4 A 6 A 2- A 3 (back left) ¾ A 1 (back right) 6 A 5 (bottom) V Figure M.50 : Rectangular solid V in xyz-space s t u S 4 S 6 S 2- S 3 (back left) ¾ S 1 (back right) 6 S 5 (bottom) W x y z V 6 A 5 (bottom)- A 3 (left) ? A 1 (back) A 2 A 6 A 4 Figure M.51 : A rectangular solid W in stu-space and the corresponding curved solid V in xyz-space Proof for Regions Parameterized by Rectangular Solids Now suppose we have a smooth change of coordinates x = x ( s, t, u ) , y = y ( s, t, u ) , z = z ( s, t, u ) . Consider a curved solid V in xyz-space corresponding to a rectangular solid W in stu-space. See Figure M.51. We suppose that the change of coordinates is one-to-one on the interior of W , and that its Jacobian determinant is positive on W . We prove the Divergence Theorem for V using the Divergence Theorem for W . Let A be the boundary of V . To prove the Divergence Theorem for V , we must show that Z A ~ F · d ~ A = Z V div ~ F dV....
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This note was uploaded on 01/06/2011 for the course CHE 6270 taught by Professor Dr.dmitry during the Spring '10 term at University of Florida.
- Spring '10