354 APPENDIX B. CURVILINEAR COORDINATES

B.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 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) .

10. Find the covariant components of velocity in spherical coordinates. Hint: v = dydt .

Now use chain rule and identify the contravariant components. Then use the tech-nique of lowering or raising index.

11. Show that v ·w = gi j (x)vi (x)v j (x) = gi j (x)vi (x)v j (x) .