What is the purpose of contraction in tensor algebra?

[joe2317]

Let Y_{1},..,Y_{k} be vector fields and let A be a tensor field of type ^{k}_{1}. Could you explain how applying k contractions to A\otimesY_{1}\otimes...Y_{k} yields A(Y_{1}...Y_{k})?

Actually, could you first explain why contraction of w\otimesY 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]

w \otimes Y has incides w_{i} \otimes Y^{j}. Contract the indices to make w \otimes Y into a scalar gives w_{i} \otimes Y^{i}. 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