Page 184 - Intro to Tensor Calculus

179

1 ∂ ∂E θ ∂B ρ 1 ∂ ∂H θ ∂D ρ
(sin θE φ ) − = − (sin θH φ ) − = J ρ +
ρ sin θ ∂θ ∂φ ∂t ρ sin θ ∂θ ∂φ ∂t
1 ∂E ρ 1 ∂ ∂B θ 1 ∂H ρ 1 ∂ ∂D θ
− (ρE φ )= − − (ρH φ )= J θ +
ρ sin θ ∂φ ρ ∂ρ ∂t ρ sin θ ∂φ ρ ∂ρ ∂t
1 ∂ 1 ∂E ρ ∂B φ 1 ∂ 1 ∂H ρ ∂D φ
(ρE θ ) − = − (ρH θ ) − = J φ +
ρ ∂ρ ρ ∂θ ∂t ρ ∂ρ ρ ∂θ ∂t

1 ∂ 2 1 ∂ 1 ∂D φ
(ρ D ρ )+ (sin θD θ )+ =%
2
ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
1 ∂ 2 1 ∂ 1 ∂B φ
(ρ B ρ )+ (sin θB θ )+ =0
2
ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
Here we have introduced the notations:
D ρ = D(1) B ρ = B(1) H ρ = H(1) J ρ = J(1) E ρ = E(1)

D θ = D(2) B θ = B(2) H θ = H(2) J θ = J(2) E θ = E(2)
D φ = D(3) B φ = B(3) H φ = H(3) J φ = J(3) E φ = E(3)

1
3
2
with x = ρ, x = θ, x = φ, h 1 =1, h 2 = ρ, h 3 = ρ sin θ
Table 5 Maxwell’s equations spherical coordinates.

Eigenvalues and Eigenvectors of Symmetric Tensors

Consider the equation
T ij A j = λA i , i, j =1, 2, 3, (2.1.19)

where T ij = T ji is symmetric, A i are the components of a vector and λ is a scalar. Any nonzero solution
A i of equation (2.1.19) is called an eigenvector of the tensor T ij and the associated scalar λ is called an
eigenvalue. When expanded these equations have the form

(T 11 − λ)A 1 + T 12 A 2 + T 13 A 3 =0
T 21 A 1 +(T 22 − λ)A 2 + T 23 A 3 =0
T 31 A 1 + T 32 A 2 +(T 33 − λ)A 3 =0.

The condition for equation (2.1.19) to have a nonzero solution A i is that the characteristic equation
should be zero. This equation is found from the determinant equation

T 11 − λ T 12 T 13

f(λ)= T 21 T 22 − λ T 23 =0, (2.1.20)

T 31 T 32
T 33 − λ