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# Engin Apps for Vector Analysis EGN 5421

USF

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This 3 page Class Notes was uploaded by April Prosacco on Wednesday September 23, 2015. The Class Notes belongs to EGN 5421 at University of South Florida taught by Arthur Snider in Fall. Since its upload, it has received 51 views. For similar materials see /class/212653/egn-5421-university-of-south-florida in General Engineering at University of South Florida.

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Date Created: 09/23/15

EGN 542l Final l2403 200 32l5 open books and notes Page l l The following field is derivable both from a vector potential and a scalar potential Give formulas for both potentials F xz yhi 2xyj 2 Evaluate the line integral h FOdR where F is the upper semicircular part the part where ygt0 of the circle x2y24 20 and F XZ y2i 2xyj The path runs from 2 O O to 2 O O Use one of the potentials you found in Problem 1 3 Let F y2i Zk i Set up but do not evaluate the flux integral of the curl VXF through the conical side surface shown below it s given parametrically by R R9r x r cos 9 y r sin 9 Z r O S 9 S 2n 0 S r S 2 ii Stokes39 theorem says this flux integral equals a line integral What is this line integral Parametrize the curve and set up but don39t evaluate the completely parametrized line integral 4 Set up but don39t evaluate the flux integrals of F y2i Zk itself and not its curl that are addressed by the divergence theorem applied to the figure below The divergence theorem for this F is easy to apply to get the net outflux what is it The volume of a cone nradius2height3 radius2 x r cos 9 22 lt gt yrsine FGN 542l Final Dec 2 l999 2 hours open book and notes l Compute a scalar and a vector potential for the two dimensional field F x4 a 6x2y2 Y4i 4x3y 4xy3j 2 Find LjF o dR for each of these fields Fi x4 a 6x2y2 y4gti My 4xy3gtj F2 xy i 2xyj zk Where C is the curve parametrized by x t2 y but use potential theory see Problem l to evaluate one of the integrals 3 Let F VHRlZ What is F What is WF 4 Let S be the sphere of radius 3 centered at the origin Let F be as in Problem 3 State the divergence theorem for the vector field F and the given sphere Then confirm the theorem by evaluating the volume integral and the surface integral The volume of a sphere is 43nR3 and its surface area is 4nR2 Each of the integrals can be evaluated by inspection 5 This problem tests your ability to set up flux and line integrals parametrically Don39t attempt to evaluate any integrals In particular they can not be evaluated by inspection The circle C of radius 3 in the z0 plane is parametrized by R9 a cos 9 i a sin 9 j O S 9 lt 2n Where the radius a is 3 The surface for which it is the boundary ie the disk S of radius 3 is parametrized by the same eguation but with the radius varying from O to 3 R9a a cos 9 i a sin 9 j O S 9 S 2n O S a S 3 or in standard notation FGN 542l Final l2299 page l Ruv v cos u i v sin u j O S u S 2n O S v S 2 Stokes39s theorem equates the flux of the curl of a vector field F through the surface with a line integral of F around the circle Let F x5j a Use the contour parametrization to set up but don t evaluate the line integral in Stokes s theorem applied to this particular S C and F Don t forget to take the curl Your final answer should look like this If 9 d9 where the integrand contains no unknowns except 9 and all the vector operations such as dot products gradients etc have been worked out b Use the surface parametrization to set up but don39t evaluate the flux integral in Stokes39s theorem applied to this particular S C and F Your final answer should look like this IIuvdudv where the integrand contains no unknowns except u and v and all the vector operations such as dot products gradients etc have been worked out c Orientation Are your 2 formulas equal or opposite negatives of each other EGN 542l Final l2299 page 2

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