Homework 8(1).pdf - Math 317 2019W2 Homework 8 Name a11 a12 a13 ~ x = A~x A = a21 a22 a23 where aij are real constants 1 Let F(~ a31 a32 a33 ~ has a

Homework 8(1).pdf - Math 317 2019W2 Homework 8 Name a11 a12...

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Math 317, 2019W2 Homework 8 Name 1. Let~F(~x) =A~x,A=a11a12a13a21a22a23a31a32a33, whereaijare real constants.(a) Find the condition onAso that~Fhas a scalar potential.(b) Find the condition onAso that~Fhas a vector potential.2. Show that the vector field~F= (x-y)bı+ (y-z)b+ (x-z)bkdoes not have a vectorpotential.3. The vector field~F= (x-y)bı+ (y-z)b+ (x-2z)bkhas a vector potential. Find avector potential~A=hA1, A2, A3iwithA3= 0.4. Leta, bR. Assume that~Fand~GareC1vector fields onR3. Prove that∇ ×(a~F+b~G) =a∇ ×~F+b∇ ×~G.5. In this exercise, we complete the computation that appears in the final step of theproof of the Stokes’ theorem forSgiven byz=f(x, y), (x, y)D, oriented upward.For a given vector field~F(x, y, z), define~G(x, y) =~F(x, y, f(x, y)), i.e.,Gi(x, y) =Fi(x, y, f(x, y)) for each 1i3. Recall that using Green’s theorem, we showed inthe lecture thatI~F·d~r=ZZ(1G2-2G1+1G3·2f-2G3·1f)dA. 6. LetSbe the part of the cylinderx2+y2= 4 that lies between the two planesz= 2

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