VECTOR CALCULUS
Use this formula to show that the area of the curved surfacex^2 +y^2 −z^2 =a^2
between the planesz=0andz=2aisπa^2(
6+
1
√
2
sinh−^12√
2
)
.
10.12 For the function
z(x, y)=(x^2 −y^2 )e−x(^2) −y 2
,
find the location(s) at which the steepest gradient occurs. What are the magnitude
and direction of that gradient? The algebra involved is easier if plane polar
coordinates are used.
10.13 Verify by direct calculation that
∇·(a×b)=b·(∇×a)−a·(∇×b).
10.14 In the following exercises,a,bandcare vector fields.
(a) Simplify
∇×a(∇·a)+a×[∇×(∇×a)] + a×∇^2 a.
(b) By explicitly writing out the terms in Cartesian coordinates, prove that
[c·(b·∇)−b·(c·∇)]a=(∇×a)·(b×c).
(c) Prove thata×(∇×a)=∇(^12 a^2 )−(a·∇)a.
10.15 Evaluate the Laplacian of the function
ψ(x, y, z)=
zx^2
x^2 +y^2 +z^2
(a) directly in Cartesian coordinates, and (b) after changing to a spherical polar
coordinate system. Verify that, as they must, the two methods give the same
result.
10.16 Verify that (10.42) is valid for each component separately whenais the Cartesian
vectorx^2 yi+xy zj+z^2 yk, by showing that each side of the equation is equal to
zi+(2x+2z)j+xk.
10.17 The (Maxwell) relationship between a time-independent magnetic fieldBand the
current densityJ(measured in SI units in A m−^2 ) producing it,
∇×B=μ 0 J,
can be applied to a long cylinder of conducting ionised gas which, in cylindrical
polar coordinates, occupies the regionρ<a.
(a) Show that a uniform current density (0,C,0) and a magnetic field (0, 0 ,B),
withBconstant (=B 0 )forρ>aandB=B(ρ)forρ<a, are consistent
with this equation. Given thatB(0) = 0 and thatBis continuous atρ=a,
obtain expressions forCandB(ρ)intermsofB 0 anda.
(b) The magnetic field can be expressed asB=∇×A,whereAis known as
the vector potential. Show that a suitableAthat has only one non-vanishing
component,Aφ(ρ), can be found, and obtain explicit expressions forAφ(ρ)
for bothρ<aandρ>a.LikeB, the vector potential is continuous atρ=a.
(c) The gas pressurep(ρ) satisfies the hydrostatic equation∇p=J×Band
vanishes at the outer wall of the cylinder. Find a general expression forp.
10.18 Evaluate the Laplacian of a vector field using two different coordinate systems
as follows.