Kronecker Delta
δ i j = { 1 i = j 0 i ≠ j \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases} δ ij = { 1 0 i = j i = j
This is the identity matrix in index notation: I i j = δ i j I_{ij} = \delta_{ij} I ij = δ ij
Properties
Contraction:
δ i k A k j = A i j \delta_{ik} A_{kj} = A_{ij} δ ik A k j = A ij
δ i k δ k j = δ i j \delta_{ik} \delta_{kj} = \delta_{ij} δ ik δ k j = δ ij
Trace:
δ i i = n \delta_{ii} = n δ ii = n
(sum over repeated index i i i n n n
Levi-Civita Symbol (3D)
ϵ i j k = { + 1 ( i , j , k ) even permutation of ( 1 , 2 , 3 ) − 1 ( i , j , k ) odd permutation of ( 1 , 2 , 3 ) 0 any index repeated \epsilon_{ijk} = \begin{cases} +1 & (i,j,k) \text{ even permutation of } (1,2,3) \\ -1 & (i,j,k) \text{ odd permutation of } (1,2,3) \\ 0 & \text{any index repeated} \end{cases} ϵ ij k = ⎩ ⎨ ⎧ + 1 − 1 0 ( i , j , k ) even permutation of ( 1 , 2 , 3 ) ( i , j , k ) odd permutation of ( 1 , 2 , 3 ) any index repeated
Even permutations: ( 1 , 2 , 3 ) (1,2,3) ( 1 , 2 , 3 ) ( 2 , 3 , 1 ) (2,3,1) ( 2 , 3 , 1 ) ( 3 , 1 , 2 ) (3,1,2) ( 3 , 1 , 2 )
Odd permutations: ( 1 , 3 , 2 ) (1,3,2) ( 1 , 3 , 2 ) ( 3 , 2 , 1 ) (3,2,1) ( 3 , 2 , 1 ) ( 2 , 1 , 3 ) (2,1,3) ( 2 , 1 , 3 )
For a 3 × 3 3 \times 3 3 × 3 A A A
det ( A ) = ∑ i , j , k = 1 3 ϵ i j k A 1 i A 2 j A 3 k \det(A) = \sum_{i,j,k=1}^{3} \epsilon_{ijk} A_{1i} A_{2j} A_{3k} det ( A ) = ∑ i , j , k = 1 3 ϵ ij k A 1 i A 2 j A 3 k
Expanding: this is a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 11 a 23 a 32 − a 12 a 21 a 33 − a 13 a 22 a 31 a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33} - a_{13}a_{22}a_{31} a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 11 a 23 a 32 − a 12 a 21 a 33 − a 13 a 22 a 31
Cross Product
Using Levi-Civita:
( a × b ) i = ∑ j , k = 1 3 ϵ i j k a j b k (\mathbf{a} \times \mathbf{b})_i = \sum_{j,k=1}^{3} \epsilon_{ijk} a_j b_k ( a × b ) i = ∑ j , k = 1 3 ϵ ij k a j b k
Or with Einstein summation convention (implied sum over repeated indices):
( a × b ) i = ϵ i j k a j b k (\mathbf{a} \times \mathbf{b})_i = \epsilon_{ijk} a_j b_k ( a × b ) i = ϵ ij k a j b k
Relationship Between δ \delta δ ϵ \epsilon ϵ
Contracted identity:
ϵ i j k ϵ i m n = δ j m δ k n − δ j n δ k m \epsilon_{ijk} \epsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} ϵ ij k ϵ imn = δ j m δ k n − δ j n δ k m
This can be verified by checking all cases.
Special cases:
ϵ i j k ϵ i j n = 2 δ k n \epsilon_{ijk} \epsilon_{ijn} = 2\delta_{kn} ϵ ij k ϵ ij n = 2 δ k n
ϵ i j k ϵ i j k = 6 \epsilon_{ijk} \epsilon_{ijk} = 6 ϵ ij k ϵ ij k = 6
Curl as Levi-Civita Expression
( ∇ × F ) i = ϵ i j k ∂ F k ∂ x j (\nabla \times \mathbf{F})_i = \epsilon_{ijk} \frac{\partial F_k}{\partial x_j} ( ∇ × F ) i = ϵ ij k ∂ x j ∂ F k
Why These Symbols Matter
Compact notation for determinants and cross products
Essential in tensor calculus and general relativity
Express vector identities elegantly (e.g., a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b ) \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) a × ( b × c ) = b ( a ⋅ c ) − c ( a ⋅ b )
Basis for understanding antisymmetric tensors
The Levi-Civita symbol encodes the “orientation” of a coordinate system — it changes sign under any swap of two indices, just like the determinant changes sign under row swaps.