13—Vector Calculus 2 41713.36 Verify the divergence theorem for the vector fieldF~=αˆxxyz+βˆy x^2 z(1 +y) +γˆz xyz^2and for the volume ( 0 < x < a), ( 0 < y < b), ( 0 < z < c).
13.37 Evaluate
∫
F~.dA~over the curved surface of the hemispherex^2 +y^2 +z^2 =R^2 andz > 0. The vectorfield is given byF~=∇×
(
αyˆx+βxˆy+γxyˆz)
. Ans:(β−α)πR^2
13.38 A vector field is specified in cylindrical coordinates to beF~ =αˆrr^2 zsin^2 θ+βθrzˆ +γˆzzrcos^2 θ. Verify
the divergence theorem for this field for the region ( 0 < r < R), ( 0 < θ < 2 π), ( 0 < z < h).
13.39 For the functionF(r,θ) =rn(A+Bcosθ+Ccos^2 θ), compute the gradient and then the divergence of
this gradient. For what values of the constantsA,B,C, and (positive or negative) integernis this last expression
zero? These coordinates are spherical. Ans: In part,n= 2,C=− 3 A,B= 0.
13.40 Repeat the preceding problem, but now interpret the coordinates as cylindrical. And don’t leave your
answers in the first form that you find them.
13.41 Evaluate the integral
∫
F~.dA~over the surface of the hemispherex^2 +y^2 +z^2 = 1withz > 0. Thevector field isF~=A(1 +x+y)xˆ+B(1 +y+z)yˆ+C(1 +z+x)ˆz. You may choose to do this problem the
hard way or the easy way. Or both. Ans:π(2A+ 2B+ 5C)/ 3
13.42 An electric field is known in cylindrical coordinates to beE~ =f(r)ˆr, and the electric charge density is
a function ofralone,ρ(r). They satisfy the Maxwell equation∇.E~ =ρ/ 0. If the charge density is given as
ρ(r) =ρ 0 e−r/r^0. ComputeE~. Explain why the behavior ofE~ is as it is for largerand for smallr.
13.43 Find a vector fieldF~ such that∇.F~=αx+βy+γand∇×F~=ˆz.
13.44 Gauss’s law says that the total charge contained inside a surface is 0
∮
E~.dA~. For the electric field of
problem10.37, evaluate this integral over a sphere of radiusR 1 > Rand centered at the origin.