Introduction
The Levi-Civita symbol (also called the permutation symbol or alternating tensor) is a mathematical object that provides a compact and elegant way to express vector operations, particularly the cross product. It is denoted by $\epsilon_{ijk}$ (or $\varepsilon_{ijk}$) and takes values of $+1$, $-1$, or $0$ depending on the permutation of its indices.
The Levi-Civita symbol $\epsilon_{ijk}$ in three dimensions is defined as:
where $i$, $j$, and $k$ each take values from the set $\{1, 2, 3\}$ (corresponding to $x$, $y$, $z$ in Cartesian coordinates).
Understanding Permutations
To understand the Levi-Civita symbol, we need to understand what even and odd permutations mean:
- An even permutation is one that can be obtained from $(1,2,3)$ by an even number of pairwise swaps. Examples: $(1,2,3)$, $(2,3,1)$, $(3,1,2)$.
- An odd permutation is one that can be obtained from $(1,2,3)$ by an odd number of pairwise swaps. Examples: $(2,1,3)$, $(3,2,1)$, $(1,3,2)$.
- If any two indices are the same, the permutation is not valid, so $\epsilon_{ijk} = 0$.
Values of the Levi-Civita Symbol
Here are all the values of $\epsilon_{ijk}$ for $i, j, k \in \{1, 2, 3\}$:
| $(i,j,k)$ | Permutation Type | $\epsilon_{ijk}$ |
|---|---|---|
| $(1,2,3)$ | Even (identity) | $+1$ |
| $(2,3,1)$ | Even (cyclic) | $+1$ |
| $(3,1,2)$ | Even (cyclic) | $+1$ |
| $(2,1,3)$ | Odd (swap 1↔2) | $-1$ |
| $(3,2,1)$ | Odd (swap 1↔3) | $-1$ |
| $(1,3,2)$ | Odd (swap 2↔3) | $-1$ |
| $(1,1,2)$, $(1,2,1)$, $(2,1,1)$, etc. | Repeated indices | $0$ |
Key Properties
- Anti-symmetric: Swapping any two indices changes the sign: $$\epsilon_{ijk} = -\epsilon_{jik} = -\epsilon_{ikj} = -\epsilon_{kji}$$
- Cyclic property: $\epsilon_{ijk} = \epsilon_{jki} = \epsilon_{kij}$ (even permutations of indices)
- Zero for repeated indices: If any two indices are equal, $\epsilon_{ijk} = 0$
- Summation identity: The product of two Levi-Civita symbols satisfies: $$\sum_{k=1}^3 \epsilon_{ijk}\epsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$$ where $\delta_{ij}$ is the Kronecker delta.
Using the Levi-Civita Symbol for Cross Products
The Levi-Civita symbol provides a compact way to express the cross product. For two vectors $\mathbf{a} = (a_1, a_2, a_3)$ and $\mathbf{b} = (b_1, b_2, b_3)$, the $i$-th component of their cross product is:
This formula gives the $i$-th component of the cross product. Using Einstein summation convention (where repeated indices are summed over), this can be written more compactly as:
where the repeated indices $j$ and $k$ are automatically summed over from 1 to 3.
Expanding the Summation
Let's see how the summation works. For the $x$-component ($i=1$), we have:
Since $\epsilon_{1jk} = 0$ unless $(j,k)$ is $(2,3)$ or $(3,2)$, and:
- $\epsilon_{123} = +1$
- $\epsilon_{132} = -1$
we get:
Similarly, for the $y$-component ($i=2$):
And for the $z$-component ($i=3$):
This matches the standard component form of the cross product!
Examples
Calculate the $x$-component of $\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = (3, 4, 0)$ and $\mathbf{b} = (2, -1, 5)$ using the Levi-Civita symbol.
Solution:
Using the formula $(\mathbf{a} \times \mathbf{b})_1 = \epsilon_{1jk} a_j b_k$:
Only terms with $\epsilon_{123}$ and $\epsilon_{132}$ are non-zero:
This matches the $x$-component from the standard cross product formula.
Calculate $\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, 5, 6)$ using the Levi-Civita symbol.
Solution:
For each component $i = 1, 2, 3$:
$x$-component ($i=1$):
$y$-component ($i=2$):
$z$-component ($i=3$):
Therefore, $\mathbf{a} \times \mathbf{b} = (-3, 6, -3)$.
Show that the Levi-Civita formula gives $\mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b}$.
Solution:
For $\mathbf{b} \times \mathbf{a}$, we have:
Using the anti-symmetry property $\epsilon_{ijk} = -\epsilon_{ikj}$:
Relabeling indices ($j \leftrightarrow k$):
This confirms the anti-commutative property of the cross product.
Use the Levi-Civita symbol to show that $\hat{x} \times \hat{y} = \hat{z}$.
Solution:
With $\hat{x} = (1, 0, 0)$ and $\hat{y} = (0, 1, 0)$, we compute the $z$-component:
The only non-zero term is when $j=1$ and $k=2$:
The $x$ and $y$ components are zero (you can verify this), so $\hat{x} \times \hat{y} = (0, 0, 1) = \hat{z}$.
Advantages of the Levi-Civita Notation
- Compact notation: The formula $(\mathbf{a} \times \mathbf{b})_i = \epsilon_{ijk} a_j b_k$ is much more compact than writing out all components explicitly.
- Generalization: The same notation works in any number of dimensions (though the cross product is only defined in 3D and 7D).
- Proofs and identities: Many vector identities are easier to prove using the Levi-Civita symbol.
- Connection to tensors: The Levi-Civita symbol is a fundamental object in tensor calculus and differential geometry.
- Computer algebra: Many symbolic computation systems use this notation internally.
Connection to Determinants
The cross product can also be written as a determinant, which is closely related to the Levi-Civita symbol:
The determinant expansion gives the same result as the Levi-Civita formula, since the determinant is essentially a sum over permutations, which is exactly what the Levi-Civita symbol encodes.