Ch14Notes

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Unformatted text preview: hat the Outward Unit Normal Vector is of the form: Let the curve C be parameterized by arc length Then our unit tangent vector for the curve C is: Let be the angle formed between and the unit vector . Then : The unit normal vector The normal vector points inward for a closed curve oriented counterclockwise So the outward unit normal vector: Summary of Ch 14 1 curtain area 2 work on open curve 3 Calculate area using a line integral 4 show field is conservative, find its potential function, use fundamental theorem of line integrals 5 work using Green’s Theorem 6 Calculate 7 Calculate 8 Calculate directly and by Gauss’s Theorem using Stoke’s Theorem directly and by Stoke’s Theorem Chapter 14 Formula Sheet Green’s Theorem (( Green’s Theorem )) OTHER VERSIONS OF GREEN’S THEOREM: (( Stoke’s Theorem in the Plane )) (( Gauss’s Theorem in the Plane )) Surface Integral: Where the surface S is defined by and is the level surface An often used surface integral: Flux of ( over the surface S ( S can be an open or closed surface ) is picked to match the direction of flow through the surface so is problem dependent ) Gauss’s Theorem or The Divergence Theorem ( is chosen to be the outward facing unit normal vector with respect to the closed surface ) Stoke’s Theorem Where the surface S is an open surface and the unit normal vector is orientation of the curve C. Use the right hand rule to match is the level surface ( depends on the to the orientation of the curve. LAW OF CONSERVATION OF ENERGY The sum of the kinetic energy and the potential energy of an object due to a CONSERVATIVE force is CONSTANT. KE + PE = Constant Therefore: Let THUS KE + PE = Constant...
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