Can the Relationship Between Levi-Civita Tensor and Kronecker Symbol Be Proven?

In summary, the conversation discusses the property of the Levi-Civita tensor and its connection to the determinant of Kronecker symbols. While this property is often taken as a given theorem in physics literature, it is not explicitly proven. The conversation also mentions that the arrangement of the sum in a determinant cannot be proven, but it is intuitively connected to the Kronecker symbol.
  • #1
raopeng
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In many physics literature I have encountered, one of the properties of Levi-Civita tensor is that [itex]ε_{ijk}ε_{lmn}[/itex]is equivalent to a determinant of Kronecker symbols. However this is only taken as a given theorem and is never proved. Is there any source which has proven this property?
 
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  • #2
Well, the tensor product is a six index object which is always expressible as a sum of a tensor product of 3 2-index objects which must necessarily be the delta Kroneckers. The nice arrangement of this sum in a determinant cannot be proven per se, just taken for granted.
 
  • #3
I can get that intuitively the Levi-Civita tensor is deeply connected to the Kronecker symbol. But if the arrangement cannot be proved per se, how does this theorem hold true...
 

1. What is the Levi-Civita Tensor product?

The Levi-Civita Tensor product, also known as the Levi-Civita symbol, is a mathematical concept used in vector calculus and differential geometry. It is a mathematical object that represents the orientation of a coordinate system in a given space.

2. How is the Levi-Civita Tensor product calculated?

The Levi-Civita Tensor product is calculated by taking the cross product of two vectors in a specific order. It follows a set of rules, such as being anti-symmetric and changing sign under interchange of any two indices.

3. What is the significance of the Levi-Civita Tensor product in physics?

The Levi-Civita Tensor product is widely used in physics, particularly in theories of electromagnetism, relativity, and quantum mechanics. It is used to describe the magnetic field and the rotation of rigid bodies, among other applications.

4. Can the Levi-Civita Tensor product be generalized to higher dimensions?

Yes, the Levi-Civita Tensor product can be generalized to higher dimensions. In three dimensions, it is represented by a 3x3 matrix, but in higher dimensions, it is represented by a multi-dimensional array.

5. How does the Levi-Civita Tensor product relate to other mathematical concepts?

The Levi-Civita Tensor product is closely related to other mathematical concepts, such as determinants, cross products, and wedge products. It is also used in the definition of the curl and divergence operations in vector calculus.

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