![]() ![]() Alternate Formulation: dS n Note that we might have to change the sign on □ □ × □ □ to match the orientation of the surface.Ĩ Surface Integrals of Vector Fields – Example 4 ![]() Let □ be a continuous vector field in □ 3 The surface integral of □ over the oriented surface S with normal n is: This is also called the Flux of F across S. Suppose the pen represents the normal vector. If you run a pen along the “inside”, suddenly it goes “outside” and then back “inside”. A famous example of a non orientable surface is the Möbius strip: take a strip of paper, twist it once, and tape the ends together. If S is a closed surface, the convention is that the positive orientation is the one where the normal vectors point outward. □ 2 If S is a smooth orientable surface described by □ □,□, then one unit normal vector is and the other is −□. We say that S is an oriented surface if we specify which normal we want to use. Then S is an orientable surface and has two unit normals at each point. □ 1 Suppose the surface S has two sides and has a tangent plane at every interior point. kg/m2) of a sheet shaped like S, then is the total mass of the sheet. Applications: If □(□,□,□) is the mass density (e.g. Let □(□,□,□) be a function defined on S. Let S be a parametric surface described by □ □,□ with (□,□) in some domain D. Presentation on theme: "Surface Integrals."- Presentation transcript:
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