## Prove that the Divergence of a Curl is Zero by using Levi Civita

Author: **Kayrol Ann B. Vacalares**

The divergence of a curl is always zero and we can prove this by using Levi-Civita symbol. The **Levi-Civita symbol**, also called the **permutation symbol** or **alternating symbol**, is a mathematical symbol used in particular in tensor calculus.

Prove that:

= 0

Proof:

Let:

and

To show that:

= 0

First,

Here are the possible values of :

if i,j,k is cyclic and non-repeating.

if i,j,k is anti-cyclic or counterclockwise.

if there are any repeated index.

Consider i,j,k to be cyclic and non-repeating, so

and

But if i is not equal to j

and if i= k

Since i,j,k is non-repeating and , therefore

Thus,

= 0

March 26th, 2012 at 8:39 am

Alexandrine…[…]Prove that the Divergence of a Curl is Zero by using Levi Civita | Quantum Science Philippines[…]…

December 12th, 2014 at 7:06 am

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September 29th, 2016 at 5:30 pm

one thing got me curious:

it is said that the levi-cevita symbol is coordinate independent, however, the way you wrote the del operator represents del in cartesian-like coordinates.

The final result is, of course, correct, but I can’t see why we don’t need to change our levi-cevita symbol (when using polar, spherical coordinates, for example)

Thanks!