Remember that the gradient is perpendicular to a

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Unformatted text preview: urface. So is a level surface. Example: Find the Flux of across S where and S: Useful ShortCut for finding . This relationship holds: is an equivalent equation just not solved for z Example: where and vs Notice: In our above example it took a little work to find Calculate but now I could use this shortcut: which is easy to do. Divide by the third component to get DONE! Section 14.7 The Divergence Theorem Gauss’s Theorem in the Plane: This gives us the FLUX outward across the closed curve C Let We know Apply Green’s Theorem to this: This is also referred to as Gauss’s Theorem. We have already derived the plane version of this from Green’s Theorem. Divergence Theorem or Gauss’s Theorem: This lifts Green’s Theorem up one dimension This gives us the FLUX outward across the closed surface Example: Find the Flux of across the closed surface S where and S: Example: Let be the closed surface octant. Find the flux of Div , , , and creating a solid tetrahedron in the first . across the closed surface where = 1+1+1=3 Example: Let be the closed surface , , and across the . closed surface where Where div creating a solid cylinder. Find the flux of Obviously cylindrical coordinates works perfectly when we have a cylinder! Section 14.8 Stoke’s Theorem Stoke’s Theorem in the Plane: This gives us the CIRCULATION along the closed curve C We are measuring the tendency for a fluid to circulate around the closed curve C. is our unit tangent vector to the curve C. We use Green’s Theorem to derive Stoke’s Theorem in the plane. Tricky Example: Suppose that the integrals taken counterclockwise around the circles are 30 and –20 respectively. Calculate and where S is the region between the circles. Stoke’s Theorem: This is a generalization of Green’s Theorem to Surfaces This gives the CIRCULATION along the open surface S = the circulation along the boundary of the surface What this implies is the above surface integral will lead to the same result regardless of the surface used as long as the boundary of the surfaces are the same! (Orientation of the boundary must be the same too!! Get opposite sign i...
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