13—Vector Calculus 2 345P
V
13.35 Work in a thermodynamic system is calculated fromdW =P dV. Assume an
ideal gas, so thatPV=nRT. (a) What is the total work,
∮
dW, done around this cycle
as the pressure increases at constant volume, then decreases at constant temperature,
finally the volume decreases at constant pressure.
(b) In the special case for which the changes in volume and pressure are very small, esti-
mate from the graph approximately what to expect for the answer. Now do an expansion
of the result of part (a) to see if it agrees with what you expect. Ans:≈∆P∆V/ 2
13.36 Verify the divergence theorem for the vector field
F~=αxyzˆx+βx^2 z(1 +y)yˆ+γxyz^2 zˆ
and 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 vector field is given byF~=∇×
(
αyxˆ+βxˆy+γxyˆz
)
. Ans:(β−α)πR^2
13.38 A vector field is given in cylindrical coordinates to beF~=rαrˆ^2 zsin^2 φ+φβrzˆ +zγzrˆ cos^2 φ.
Verify the divergence theorem for this field for the region ( 0 < r < R), ( 0 < φ < 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, negative, or zero)
integernis this result,∇.∇F, zero? These coordinates are spherical, and this combination div grad
is the Laplacian.
Ans: In part,n= 2,C=− 3 A,B= 0.
13.40 Repeat the preceding problem, but now interpret the coordinates as cylindrical (changeθtoφ).
And don’t necessarily 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.
The vector field isF~=A(1 +x+y)xˆ+B(1 +y+z)yˆ+C(1 +z+x)zˆ. 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 asρ(r) =ρ 0 e−r/r^0. ComputeE~. Demonstrate the behavior ofE~ is for largerand
for smallr.
13.43 Repeat the preceding problem, but nowris a spherical coordinate.
13.44 Find a vector fieldF~such that∇.F~=αx+βy+γand∇×F~ =ˆz. Next, find an infinite
number of such fields.
13.45 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.
13.46 (a) In cylindrical coordinates, for whatndoes the vector field~v=rnφˆhave curl equal to zero?
Draw it.