**Roel N. Baybayon**

MSPhysics1

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We are going to prove the following vector identity using Levi-Civita symbol:

Solution:

Let , , , .

Then,

By definition:

We have to let m=n so that,

Levi-Civita symbol can be expressed in terms of Kronecker delta given by:

Thus,

Posted in Electrodynamics, Quantum Science Philippines **|** 1 Comment »

By **Sim P. Bantayan**, MS Physics I, MSU-IIT

Let ,

and where and .

**1. Prove that .**

Proof:

Now, . Since* i*=*j* for the divergence of normal unit vector **n, **

** **

but (*i*=*j)*. Moreover, for three dimensions, , so

Therefore, .

**2. Prove that .**

Proof:

. Since *i*=*j* for the curl of normal unit vector **n,**

but (index *i* is repeated).

Therefore, .

Posted in Electrodynamics **|** No Comments »

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

Posted in Electrodynamics **|** 3 Comments »

**Applying **** and **** in Proving the Vector Formula: **

*By: Quennie J. Paylaga*

Prove:

using Kronecker Delta Function and Levi-Civita Symbol.

To prove this, we let

We can write the expression for in summation form as:

*where *

*where i, j, l are dummy summation variables. Each of which can be any letter (a,b,c) or number (1,2,3).*

In the same way, we can write as:

Thus, we have prove that

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