What is the purpose of contraction in tensor algebra?

[joe2317]

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.

 

[HallsofIvy]

Isn't that pretty much the definition of "contraction"?

 

[joe2317]

Could you explain in more detail?
Definition of contraction can be found here:http://en.wikipedia.org/wiki/Tensor_contraction
Thanks.

 

[AlphaNumeric2]

[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.

 

[quetzalcoatl9]

a special case is the dot product of two vectors, this is how everyone really things about contraction anyway