PHZ 3113 Fall 2010
Homework #4, Due Monday, October 4
1. Define elliptical coordinates u, v by x =
√
u2 + c2 cos v,
y = u sin v .
(a) Write the area element dA = dx dy in terms of elliptical u, v.
(b) Using elliptical coordinates, compute the area of an ellipse with semimajor axis a, semiminor axis b, where a2 = b2 + c2 .
2
1/3
x − y2
2. Let the scalar field φ be given by φ(x, y) =
.
x2 + y 2
(a) Compute the components of the gradient ∇φ. Where is the derivative troublesome?
(b) Compute the directional derivative of φ in the direction n̂ = x̂ cos α + ŷ sin α. Write
your answer in terms of polar coordinates ρ, φ, where x = ρ cos φ, y = ρ sin φ. What is the
directional derivative along a radial ray (fixed φ) as ρ → 0?
3. Let the vector field v be
v=
where r =
p
x2 + y 2 + z 2 .
y(y 2 + z 2 )
x(x2 + z 2 )
xyz
x̂
+
ŷ − 3 ẑ ,
3
3
r
r
r
(a) Compute ∇ · v. Compute ∇ × v.
(b) Compute ∇(∇ · v) and ∇2 v. Compute ∇ × (∇ × v) two different ways.
4. In analogy with electrostatics, where the electric field E can be obtained from an electrostatic potential E = −∇V where V satisfies Poisson’s equation ∇2 V = −ρe /0 , the
gravitational field g can be obtained from a gravitational potential g = −∇φ that satisfies
∇2 φ = 4πGρg . In numerical simulations of large-scale cosmological structure, it is sometimes
useful to “soften” the Newtonian gravitational potential φg = −GM/r of a point particle of
mass M as
GM
φa = −
.
r+a
(a) What is the gravitational field g derived from φa ? How does g behave as r → ∞? Show
that the softening introduces a maximum on the gravitational force between particles.
(b) The softened φa can be interpreted as the potential of a slightlyR fuzzy particle with a
mass density ρa (r). What is ρa (r)? What is the total mass, Ma = d3 v ρa ? What radius
r1/2 contains half the mass?