Understand Vector Product with Levi Civita Symbols

In summary, the Levi Civita symbol is used in the vector product, and the product of two Levi Civita symbols can be expressed as Kronecker Deltas. The dot product of two vector products can also be written in terms of the Levi Civita symbol, and by substituting the values of the vector products, the final expression is obtained. The proof for these concepts can be found at the provided URL.
  • #1
Septim
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Hello I am a graduate student and it is my first year in physics. Our instructor showed vector product with Levi Civita symbol and I did not understand the part where the product of two Levi Civita symbols is expressed as Kronecker Deltas. Additionally I did not get the proof on the URL http://folk.uio.no/patricg/teaching/a112/levi-civita/ can you explain these. Thanks for your contribution.
 
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  • #2
img8.png


now take dot product of two vector products
(aXb).(cXd), which can be expressed further as
(aXb).(cXd)=c.((d.b)a - (d.a)b)

plug in the values of vector product using above equation(which includes levi civita symbols)
and then finally this would lead to the expression
img19.png
 

1. What is the Levi Civita symbol in vector product?

The Levi Civita symbol, also known as the permutation symbol, is a mathematical notation used to represent the sign of a permutation of a set of numbers. In vector product, it is used to determine the direction of the resultant vector.

2. How is the Levi Civita symbol used in vector product?

In vector product, the Levi Civita symbol is used to determine the direction of the resultant vector by assigning a sign (+ or -) to each permutation of the components of the two vectors being multiplied. It is also used to simplify the calculation of the cross product.

3. What is the significance of the Levi Civita symbol in vector product?

The Levi Civita symbol is significant in vector product as it allows for a concise representation of the direction of the resultant vector. It is also useful in simplifying calculations and has applications in various fields such as physics and engineering.

4. How does the Levi Civita symbol handle three-dimensional vector products?

In three-dimensional vector product, the Levi Civita symbol is represented as a 3x3 matrix. Each row of the matrix represents a component of the first vector, while each column represents a component of the second vector. The sign of each permutation can then be determined by the matrix elements.

5. Are there any other applications of the Levi Civita symbol?

Yes, the Levi Civita symbol is used in various areas of mathematics and physics, such as in differential geometry, electromagnetism, and quantum mechanics. It is also used in the study of rotations and angular momentum in classical mechanics.

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