What is the purpose of contraction in tensor algebra?

In summary, when applying k contractions to a tensor field A of type ^{k}_{1} and vector fields Y_{1},...,Y_{k}, we end up with A(Y_{1}...Y_{k}). This is because contraction of a 1-form w and a vector field Y is defined as w(Y), which is equivalent to contracting the indices of w \otimes Y^{j} to make it a scalar. This same principle applies to all other cases, such as the dot product of two vectors. The definition of contraction can be found here: http://en.wikipedia.org/wiki/Tensor_contraction.
  • #1
joe2317
6
0
Let Y[tex]_{1}[/tex],..,Y[tex]_{k}[/tex] be vector fields and let A be a tensor field of type [tex]^{k}_{1}[/tex]. Could you explain how applying k contractions to A[tex]\otimes[/tex]Y[tex]_{1}[/tex][tex]\otimes[/tex]...Y[tex]_{k}[/tex] yields A(Y[tex]_{1}[/tex]...Y[tex]_{k}[/tex])?

Actually, could you first explain why contraction of w[tex]\otimes[/tex]Y is equal to w(Y)?
Here, w is a 1-form and Y is a vector field.
Thank you.
 
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  • #2
Isn't that pretty much the definition of "contraction"?
 
  • #4
[tex]w \otimes Y[/tex] has incides [tex]w_{i} \otimes Y^{j}[/tex]. Contract the indices to make [tex]w \otimes Y[/tex] into a scalar gives [tex]w_{i} \otimes Y^{i}[/tex]. This is the definition of w(Y).

Similarly for everything else.
 
  • #5
a special case is the dot product of two vectors, this is how everyone really things about contraction anyway
 

1. What is a tensor?

A tensor is a mathematical object that describes the relationship between different vectors and scalars in a multi-dimensional space. It is used in various fields such as physics, engineering, and computer science.

2. What is the contraction of a tensor?

The contraction of a tensor is a mathematical operation that involves summing over one or more indices in a tensor. It results in a new tensor with fewer indices, representing a more simplified relationship between the vectors and scalars.

3. How is the contraction of a tensor performed?

The contraction of a tensor is performed by multiplying the elements of the tensor with the corresponding elements of a specified vector or scalar, and then summing over the repeated indices.

4. What is the significance of the contraction of a tensor?

The contraction of a tensor is significant because it allows for the simplification of complex mathematical expressions, making them easier to understand and manipulate. It is also a useful tool in solving problems in fields such as mechanics, relativity, and electromagnetism.

5. Can the contraction of a tensor be applied to tensors of any order?

Yes, the contraction of a tensor can be applied to tensors of any order, as long as the indices being contracted are repeated indices. The result will always be a new tensor with fewer indices than the original.

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