A.2 Levi-Civita Antisymmetric Tensor
197
ε ij k ε pj k = δ ip δ jj − δ ij δ jp = 3δ ip − δip = 2δ ip ,
(7.134)
ε ij k ε ij k = 2δ ii = 6.
(7.135)
These Eqs. (7.133), (7.134), (7.135), particularly Eq. (7.133), are extensively
utilized in manipulating vector formulas. For example, the vector triple product is
rearranged as follows,
[A × (B × C)] i = ε ij k A j [B × C] k = ε ij k A j (ε klm B l C m )
= ε ij k ε lmk A j B l C m = (δ il δ jm − δ im δ jl )A j B l C m = A j B i C j − A j B j C i
= B i (A · C) − C i (A · B),
where we have employed the permutation relation, ε klm = ε lmk , and Eq. (7.133) in
the above derivation.
[Problem 7.4] Derive the following formulas (i)–(vi) using the Levi-Civita tensor
(see Sect. 7.6.4). (A, B, C, D refer to vectors, and φ to a scalar.)
(i) (A × B) · (C × D) = (A · C) (B · D) − (A · D) (B · C)
(ii) ∇ × (∇ × A) = ∇ (∇ · A) − ∇
2 A
(iii) ∇ · (∇ × A) = 0
(iv) ∇ × (∇φ) = 0
(v) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
(vi) ∇ × (A × B) = A (∇ · B) + (B · ∇) A − B (∇ · A) − (A · ∇) B
The Levi-Civita tensor is invariant under rotation of the coordinates,
D ip D jq D kr ε pqr = ε ij k ,
(7.137)
where D is the rotational matrix in Eq. (3.43). This feature will be utilized in
Chap. 8.
Proof Consider a rotational matrix D that converts a set of orthonormal vectors
{e x , e y , e z } to {e x , e y , e z },
⎛
⎝
e x
e y
e z
⎞
⎠ = D
⎛
⎝
e x
e y
e z
⎞
⎠ .
197
ε ij k ε pj k = δ ip δ jj − δ ij δ jp = 3δ ip − δip = 2δ ip ,
(7.134)
ε ij k ε ij k = 2δ ii = 6.
(7.135)
These Eqs. (7.133), (7.134), (7.135), particularly Eq. (7.133), are extensively
utilized in manipulating vector formulas. For example, the vector triple product is
rearranged as follows,
[A × (B × C)] i = ε ij k A j [B × C] k = ε ij k A j (ε klm B l C m )
= ε ij k ε lmk A j B l C m = (δ il δ jm − δ im δ jl )A j B l C m = A j B i C j − A j B j C i
= B i (A · C) − C i (A · B),
where we have employed the permutation relation, ε klm = ε lmk , and Eq. (7.133) in
the above derivation.
[Problem 7.4] Derive the following formulas (i)–(vi) using the Levi-Civita tensor
(see Sect. 7.6.4). (A, B, C, D refer to vectors, and φ to a scalar.)
(i) (A × B) · (C × D) = (A · C) (B · D) − (A · D) (B · C)
(ii) ∇ × (∇ × A) = ∇ (∇ · A) − ∇
2 A
(iii) ∇ · (∇ × A) = 0
(iv) ∇ × (∇φ) = 0
(v) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
(vi) ∇ × (A × B) = A (∇ · B) + (B · ∇) A − B (∇ · A) − (A · ∇) B
The Levi-Civita tensor is invariant under rotation of the coordinates,
D ip D jq D kr ε pqr = ε ij k ,
(7.137)
where D is the rotational matrix in Eq. (3.43). This feature will be utilized in
Chap. 8.
Proof Consider a rotational matrix D that converts a set of orthonormal vectors
{e x , e y , e z } to {e x , e y , e z },
⎛
⎝
e x
e y
e z
⎞
⎠ = D
⎛
⎝
e x
e y
e z
⎞
⎠ .