Page 183 - Intro to Tensor Calculus

178

∂E z ∂E y ∂B x ∂H z ∂H y ∂D x
− = − − = J x + ∂D x ∂D y ∂D z
∂y ∂z ∂t ∂y ∂z ∂t + + = %
∂x ∂y ∂z
∂E x ∂E z ∂B y ∂H x ∂H z ∂D y
− = − − = J y +
∂z ∂x ∂t ∂z ∂x ∂t
∂B x ∂B y ∂B z
∂E y ∂E x ∂B z ∂H y ∂H x ∂D z + + =0
− = − − = J z + ∂x ∂y ∂z
∂x ∂y ∂t ∂x ∂y ∂t
Here we have introduced the notations:

D x = D(1) B x = B(1) H x = H(1) J x = J(1) E x = E(1)
D y = D(2) B y = B(2) H y = H(2) J y = J(2) E y = E(2)
D z = D(3) B z = B(3) H z = H(3) J z = J(3) E z = E(3)

3
2
1
with x = x, x = y, x = z, h 1 = h 2 = h 3 =1
Table 3 Maxwell’s equations Cartesian coordinates

1 ∂E z ∂E θ ∂B r 1 ∂H z ∂H θ ∂D r
− = − − = J r +
r ∂θ ∂z ∂t r ∂θ ∂z ∂t
∂E r ∂E z ∂B θ ∂H r ∂H z ∂D θ
− = − − = J θ +
∂z ∂r ∂t ∂z ∂r ∂t
1 ∂ 1 ∂E r ∂B z 1 ∂ 1 ∂H r ∂D z
(rE θ ) − = − (rH θ ) − = J z +
r ∂r r ∂θ ∂t r ∂r r ∂θ ∂t
1 ∂ 1 ∂D θ ∂D z 1 ∂ 1 ∂B θ ∂B z
(rD r )+ + = % (rB r )+ + =0
r ∂r r ∂θ ∂z r ∂r r ∂θ ∂z

Here we have introduced the notations:

D r = D(1) B r = B(1) H r = H(1) J r = J(1) E r = E(1)
D θ = D(2) B θ = B(2) H θ = H(2) J θ = J(2) E θ = E(2)

D z = D(3) B z = B(3) H z = H(3) J z = J(3) E z = E(3)
1
2
3
with x = r, x = θ, x = z, h 1 =1, h 2 = r, h 3 =1.
Table 4 Maxwell’s equations in cylindrical coordinates.