Levi-Civita permutation symbol

Levi-Civita permutation symbol

Definition 1

.

Let

k

i

∈

{

1

,

⋯

,

n

}

for all

i

=

1

,

⋯

,

n

.
The

Levi-Civita permutation symbols

ε

k

1

⁢

⋯

⁢

k

n

and

ε

k

1

⁢

⋯

⁢

k

n

are
defined as

ε

k

1

⁢

⋯

⁢

k

m

=

ε

k

1

⁢

⋯

⁢

k

m

=

{

+

1

𝑤ℎ𝑒𝑛

{

l

↦

k

l

}

is an even permutation (of

{

1

,

⋯

,

n

}

),

-

1

𝑤ℎ𝑒𝑛

{

l

↦

k

l

}

is an odd permutation,

0

otherwise, i.e., when

⁢

k

i

=

k

j

,

for some

⁢

i

≠

j

.

The Levi-Civita permutation symbol is a special case of the

generalized
Kronecker delta symbol

. Using this fact one can write the Levi-Civita permutation
symbol as the

determinant

of an

n

×

n

matrix consisting of traditional
delta symbols. See the entry on the generalized

Kronecker symbol

for details.

When using the Levi-Civita permutation symbol and the generalized Kronecker delta
symbol, the

Einstein summation convention

is usually employed. In the below,
we shall also use this convention.

Properties

•

When

n

=

2

, we have for all

i

,

j

,

m

,

n

in

{

1

,

2

}

,

ε

i

⁢

j

⁢

ε

m

⁢

n

=

δ

i

m

⁢

δ

j

n

-

δ

i

n

⁢

δ

j

m

,

(1)

ε

i

⁢

j

⁢

ε

i

⁢

n

=

δ

j

n

,

(2)

ε

i

⁢

j

⁢

ε

i

⁢

j

=

2

.

(3)

•

When

n

=

3

, we have for all

i

,

j

,

k

,

m

,

n

in

{

1

,

2

,

3

}

,

ε

j

⁢

m

⁢

n

⁢

ε

i

⁢

m

⁢

n

=

2

⁢

δ

j

i

,

(4)

ε

i

⁢

j

⁢

k

⁢

ε

i

⁢

j

⁢

k

=

6

.

(5)

Let us prove these properties. The proofs are instructional since they
demonstrate typical argumentation methods for manipulating the

permutation symbols

.

Proof.

For equation

1

, let us first note that both sides
are

antisymmetric

with respect of

i

⁢

j

and

m

⁢

n

. We therefore only need
to consider the case

i

≠

j

and

m

≠

n

. By substitution, we see that
the equation holds for

ε

12

⁢

ε

12

, i.e., for

i

=

m

=

1

and

j

=

n

=

2

. (Both sides are then one). Since the equation is
anti-symmetric in

i

⁢

j

and

m

⁢

n

, any set of values for these can be
reduced the above case (which holds). The equation
thus holds for all values of

i

⁢

j

and

m

⁢

n

.
Using equation

1

, we have for equation

2

ε

i

⁢

j

⁢

ε

i

⁢

n

=

δ

i

i

⁢

δ

j

n

-

δ

i

n

⁢

δ

j

i

=

2

⁢

δ

j

n

-

δ

j

n

=

δ

j

n

.

Here we used the Einstein summation convention with

i

going from

1

to

2

.
Equation

3

follows similarly from equation

2

.
To establish equation

4

, let us first observe that both sides
vanish when

i

≠

j

. Indeed, if

i

≠

j

, then one can not choose

m

and

n

such that both permutation symbols on the left are nonzero. Then,
with

i

=

j

fixed, there are only two ways to choose

m

and

n

from the remaining
two indices. For any such indices, we have

ε

j

⁢

m

⁢

n

⁢

ε

i

⁢

m

⁢

n

=

(

ε

i

⁢

m

⁢

n

)

2

=

1

(no

summation

),
and the result follows. The last property follows since

3

!

=

6

and for any
distinct indices

i

,

j

,

k

in

{

1

,

2

,

3

}

, we have

ε

i

⁢

j

⁢

k

⁢

ε

i

⁢

j

⁢

k

=

1

(no summation).

□

Examples and Applications.

•

The determinant of an

n

×

n

matrix

A

=

(

a

i

⁢

j

)

can be written
as

det

⁡

A

=

ε

i

1

⁢

⋯

⁢

i

n

⁢

a

1

⁢

i

1

⁢

⋯

⁢

a

n

⁢

i

n

,

where each

i

l

should be summed over

1

,

…

,

n

.

•

If

A

=

(

A

1

,

A

2

,

A

3

)

and

B

=

(

B

1

,

B

2

,

B

3

)

are vectors in

ℝ

3

(represented in some right hand oriented

orthonormal basis

), then
the

i

th

component

of their

cross product

equals

(

A

×

B

)

i

=

ε

i

⁢

j

⁢

k

⁢

A

j

⁢

B

k

.

For instance, the first component of

A

×

B

is

A

2

⁢

B

3

-

A

3

⁢

B

2

. From the above expression for the cross product,
it is clear that

A

×

B

=

-

B

×

A

.
Further, if

C

=

(

C

1

,

C

2

,

C

3

)

is a vector like

A

and

B

, then
the

triple scalar product

equals

A

⋅

(

B

×

C

)

=

ε

i

⁢

j

⁢

k

⁢

A

i

⁢

B

j

⁢

C

k

.

From this expression, it can be seen that the triple scalar product is
antisymmetric when exchanging any

adjacent

arguments.
For example,

A

⋅

(

B

×

C

)

=

-

B

⋅

(

A

×

C

)

.

•

Suppose

F

=

(

F

1

,

F

2

,

F

3

)

is a

vector field

defined on some
open set of

ℝ

3

with

Cartesian coordinates

x

=

(

x

1

,

x

2

,

x

3

)

. Then
the

i

th component of the curl of

F

equals

(

∇

×

F

)

i

⁢

(

x

)

=

ε

i

⁢

j

⁢

k

⁢

∂

∂

⁡

x

j

⁢

F

k

⁢

(

x

)

.

Title

Levi-Civita permutation symbol

Canonical name

LeviCivitaPermutationSymbol

Date of creation

2013-03-22 13:31:29

Last modified on

2013-03-22 13:31:29

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

13

Author

matte (1858)

Entry type

Definition

Classification

msc 05A10

Related topic

KroneckerDelta

Related topic

GeneralizedKroneckerDeltaSymbol

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