Trying to understand Levi-Civita Symbol and notation

In summary, the Levi-Civita symbol is a mathematical symbol that represents the number of permutations of a set of indices (i,j,k) and has values of -1, 1, or 0 depending on the number of permutations. It can also be written as a sum in higher dimensions, with i ranging from 1 to n and the values of j and k being fixed. Its product with another Levi-Civita symbol can be calculated using a determinant.
  • #1
Werbel22
8
0
Hello,

I am having a little difficulty understanding what exactly the Levi-Civita symbol is about.

In the past I believed that it was equal to 1, -1 and 0, depending on the number of permutations of i,j,k. I had just accepted that to be the extent of it.

However, now I am seeing things like summations with i,j,k mentioned as WELL as other letters, for example m and n in the link below. I tried reading online about it to understand what exactly it is, especially here:

http://en.wikipedia.org/wiki/Levi-Civita_symbol#Relation_to_Kronecker_delta

How can something that is the value of -1, 1 or 0 be written as a sum, with i=1 up to 3 as written in the link, without any mention of the values of j or k? Whenever I see summation, I think substitute i=1 first, then add with i=2, then add the value with i=3, and that's the answer. But if it's E_ijk E_imn, if I sub into the 'contracted epsilon identity' as shown on wikipedia I get

E_1jk E_1mn + E2jk E_2mn + E3jk E_3mn

How do I know what to do with the levi-civita symbol if I don't know what to with all these other letters?

What does this expression even mean? :S

Sorry for my lack of knowledge here, I haven't had much experience ever using it and I'm worried I'll fall behind in class if I don't get this cleared up.

Thanks!
 
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  • #2
Werbel22 said:
… However, now I am seeing things like summations with i,j,k mentioned as WELL as other letters, for example m and n in the link below. I tried reading online about it to understand what exactly it is, especially here:

http://en.wikipedia.org/wiki/Levi-Civita_symbol#Relation_to_Kronecker_delta

How can something that is the value of -1, 1 or 0 be written as a sum, with i=1 up to 3 as written in the link, without any mention of the values of j or k? Whenever I see summation, I think substitute i=1 first, then add with i=2, then add the value with i=3, and that's the answer. …

Hello Werbel22! :smile:

i=1 up to 3 is only for three dimensions.

For the generalisation to n dimensions, i = 1 to n.

The product of two Levi-Civita symbols in n dimensions is given later on that page, as a determinant.

The ∑ with i=1 up to 3 as written in the link is for fixed values of j k m and n (so the RHS is a function of j k m and n) … there's no summing over j and k because they're fixed. :wink:
 
  • #3
So then what is the value of the levi-civita symbol then? I only know it to be -1, 1, 0, and I thought you need to know about j and k as well to find it's value?
 
  • #4
?? :confused: if j and k are given, then you do know their values.
 
  • #5


Dear questioner,

The Levi-Civita symbol (ε) is a mathematical symbol used in vector calculus and tensor analysis. It represents a mathematical object called a permutation symbol, which is used to define the cross product of two vectors in three-dimensional space. The symbol is defined as follows:

εijk = 1 if (i,j,k) is an even permutation of (1,2,3)
εijk = -1 if (i,j,k) is an odd permutation of (1,2,3)
εijk = 0 if any two indices are equal

In other words, the Levi-Civita symbol assigns a value of either 1, -1, or 0 to each possible permutation of the indices i, j, and k.

Now, regarding the notation you mentioned, the expression E_ijk E_imn is known as a contraction of the Levi-Civita symbol. This means that the symbol is being multiplied by itself with different indices, and then summed over all possible values of those indices. In this case, the indices i and m are being summed over, while j and k are not mentioned. This is because the Levi-Civita symbol is antisymmetric, meaning that it changes sign when any two indices are swapped. Therefore, when you expand the expression using the definition of the symbol, you will get terms with different permutations of the indices, and the ones with the same permutation (such as E_1jk E_1mn) will cancel out.

In general, the Levi-Civita symbol is used to simplify expressions involving tensors, and it follows certain rules and properties which make it a useful tool in vector and tensor analysis. I suggest consulting a textbook or seeking guidance from your instructor to fully understand its applications and uses. Do not worry about falling behind in class, as it takes time and practice to fully grasp the concepts involved in this notation.

I hope this explanation helps clarify your understanding of the Levi-Civita symbol and its notation. Keep studying and don't be afraid to ask for help when needed. Best of luck in your studies!
 

1. What is the Levi-Civita Symbol?

The Levi-Civita Symbol, also known as the permutation symbol, is a mathematical symbol used to represent the sign of a permutation of a set of numbers. It is often denoted by the Greek letter epsilon (ε) and is defined as 1 when the permutation is even, -1 when the permutation is odd, and 0 when any two elements are equal.

2. What is the purpose of the Levi-Civita Symbol?

The Levi-Civita Symbol is used in various branches of mathematics, including vector calculus, differential geometry, and physics. It is particularly useful in cross and dot product calculations, determining the orientation of vectors, and expressing the cross product of two vectors in terms of determinants.

3. How is the Levi-Civita Symbol notated?

The Levi-Civita Symbol is usually notated using a superscript epsilon (ε) with three indices in either upper or lower positions. For example, εijk or εijk. The indices represent the order of the elements in a permutation, and the symbol is read as "epsilon i, j, k".

4. What is the difference between the Levi-Civita Symbol and the Kronecker Delta?

The Levi-Civita Symbol and the Kronecker Delta (δ) are both used to represent certain mathematical concepts involving permutations and determinants. However, the Levi-Civita Symbol is an alternating symbol, meaning its value changes based on the order of the elements, while the Kronecker Delta is a constant symbol with a value of 1 when the indices are equal and 0 when they are not.

5. How can I use the Levi-Civita Symbol in calculations?

The Levi-Civita Symbol can be used in various mathematical calculations, such as computing the determinant of a matrix, finding the cross product of two vectors, and solving vector equations. It is important to follow the rules of index notation and the properties of the symbol, such as its antisymmetry and permutation properties, to correctly apply it in calculations.

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