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$$\otimes$$Y$$_{1}$$$$\otimes$$...Y$$_{k}$$ yields A(Y$$_{1}$$...Y$$_{k}$$)?

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

$$w \otimes Y$$ 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