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# Calculus III MATH 241

UT

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This 2 page Class Notes was uploaded by John MacGyver on Monday October 26, 2015. The Class Notes belongs to MATH 241 at University of Tennessee - Knoxville taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/229828/math-241-university-of-tennessee-knoxville in Mathematics (M) at University of Tennessee - Knoxville.

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Date Created: 10/26/15

Math 241 Spring 2006 NAME List 8 Divergence and curl of vector elds7 theorems of Stokes and Gauss7 physics applications 1 Transform the ux of the curl of the given vector eld on the given surface to a line integral using Stokes7 theorem then compute the integral iFIyyyz yawn 5 upper unit hemisphere oriented by the upward normal iiFIyyyz y 21 S 2 2 0 portion of the paraboloid 2 l 7 12 7 y2 oriented by the upward normal iiiFIyyyz 127 7y712y Sztetrahedron bounded by the coordinate planes and the plane 31 y 32 6 except for the face on the 12 plane oriented by the outward norma 2 Use Stokes7 theorem to show the line integrals have the values given specify how C must be oriented i C ydz zdy zdz TraQS where C is the intersection of the sphere 12y222 a2 and the planezyz0 ii f0 y zdz 2 zdy I ydz 0 where C is the intersection of the cylinder 12 y2 2y and the plane y 2 iii Cy 7 zdz 2 7 zdy I 7 ydz 27raa b where C is the intersection of the cylinder 12 y2 a2 and the plane za 212 l where a gt 012 gt 0 3 Let F 7y12 y2z12 y2z de ned on the torus 5 obtained by rotating the circle I 7 2 2 l in the 12 plane about the 2 axis Show that the curl of F is zero on S but the line integral of F along the circle 12 y2 9 2 0 is not zero Why doesnlt this contradict Stokes7 theorem 4 For each F given below can there be a vector eld G so that Fcurl G zy2y22 212 yzzy2 5 Prove the following vector identities for arbitrary functions u vector elds F divuF udivF Vu F curluF ucurlF Vu X F divF X G curlF G 7 F curlG curl curl F VdivF 7 AF 6 Use the divergence theorem to compute the outward ux of F across the boundary of the given region r 127 72zy7312 D 12 y2 22 S 4 rst octant 7 177 y77 2 D l S 7 2 S 4 y71y77z D 12 y2 S 40 S 2 S 12 y2i 7 Use the divergence theorem to show that the volume of the region D bounded by a closed surface S in R3 equals onethird of the ux of FXyz across 5 8i Use the divergence theorem to evaluate 21 2y 22dA7 S where S is the sphere 12 y2 22 1 ii Use the divergence theorem to evaluate the ux of the vector eld given below on the upper unit hemisphere with upward normal orientation7 by trans ferring it to an integral over the unit disk in the plane 2 0 F y22713 tan2z2 y2i 9 Conservation of mass Let V be the velocity vector eld of a uid with density pI y z t ttime in a region T bounded by a closed surface 5 Show that the conservation of mass equation awayom is equivalent to the continuity equation divpv g 0 10 Maxwell s equations and the wave equation Maxwellls equations for timedependent electric eld E and magnetic eld B in vacuum read 8B 8E div E 07 div B 07 curl E 7 curl B sono i Use the equations and the identities in problem 5 to show each component of E and of B satis es the wave equation with speed of propagation c 60u0 12 um 7 C2Au 0i

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