Mathematical Physics Vol 1

Chapter 4. Field theory

216

Solution

a) e r = sin θ cos φ i + sin θ sin φ j + cos θ k , e θ = cos θ cos φ i + cos θ sin φ j − sin θ k , e φ = − sin φ i + cos φ j , b) i = sin θ cos φ e r + cos θ cos φ e θ − sin φ e φ , j = sin θ sin φ e r + cos θ sin φ e θ + cos φ e φ , k = cos θ e r − sin θ e θ .

Exercise 173 Express the vector A = 2 y i − z j + 3 x k in spherical coordinates and determine A ρ , A φ , A z .

Solution

A = A r e r + A θ e θ + A φ e φ ,

gde je

2 θ sin φ cos φ − r sin θ cos θ sin φ + 3 r sin θ cos θ cos φ ,

A r = 2 r sin

2 θ sin φ − 3 r 2 sin 2 θ cos φ ,

A θ = 2 r sin θ cos θ sin φ cos φ − r cos

2 φ − r cos θ cos φ .

A φ = − 2 r sin θ sin

Exercise 174 Prove that the following coordinate systems are orthogonal

a) parabolic cylindrical, b) elliptic cylindrical, and c) spheroid.

Exercise 175 Prove that a curvilinear coordinate system is orthogonal iff g pq = 0 for p̸ = q .

Exercise 176 Find the Jacobian J =

∂ ( x , y , z ) ∂ ( u 1 , u 2 , u 3 )

for the following curvilinear systems

a) cylindrical, b) spherical, c) parabolic cylindrical, d) elliptic cylindrical, e) spheroid.