25.8. EXERCISES 481

25.8 Exercises1. Let y1 = x1 +2x2,y2 = x2 +3x3,y3 = x1 + x3. Let

F (x) = x1e1 (x)+ x2e2 (x)+(x3)2

e(x) .

Find div(F )(x) .

2. For the coordinates of the preceding problem, and φ a scalar field, find

(∇φ (x))3

in terms of the partial derivatives of φ taken with respect to the variables xi.

3. Let y1 = 7x1+2x2,y2 = x2+3x3,y3 = x1+x3. Let φ be a scalar field. Find ∇2φ (x) .

4. Derive ∇2u in cylindrical coordinates, r,θ ,z, where u is a scalar field on R3.

x = r cosθ , y = r sinθ , z = z.

5. ↑ Find all solutions to ∇2u = 0 which depend only on r where r ≡

√x2 + y2.

6. Derive ∇2u in spherical coordinates.

7. ↑Let u be a scalar field on R3. Find all solutions to ∇2u = 0 which depend only on

ρ ≡√

x2 + y2 + z2.

8. The temperature, u, in a solid satisfies ∇2u = 0 after a long time. Suppose in a long

pipe of inner radius 9 and outer radius 10 the exterior surface is held at 100◦ whilethe inner surface is held at 200◦ find the temperature in the solid part of the pipe.

9. Show {li j

}=

∂ei

∂x j ·el .

Find the Christoffel symbols of the second kind for spherical coordinates in whichx1 = φ , x2 = θ , and x3 = ρ. Do the same for cylindrical coordinates letting x1 = r,x2 = θ , x3 = z.

10. Show velocity can be expressed as v = vi (x)ei (x) , where

vi (x) =∂ ri

∂x jdx j

dt− rp (x)

{pik

}dxk

dt

and ri (x) are the covariant components of the displacement vector,

r = ri (x)ei (x) .

11. ↑ Using problem 9 and 10, show the covariant components of velocity in sphericalcoordinates are

v1 = ρ2 dφ

dt, v2 = ρ

2 sin2 (φ)dθ

dt, v3 =

dρ

dt.

Hint: First observe that if r is the position vector from the origin, then r = ρe3 sor1 = 0 = r2, and r3 = ρ. Now use 10.