Can Rank Two Tensors Be Formed Using Dot Products?

[xGAME-OVERx]

Hi All,

I'm currently doing undergraduate research involving a lot of work with rank two Cartesian tensors, and I'm having trouble finding much information or good references on the foundations of such things.

It's my understanding that a rank two tensor can be written $$T = T_{ij} \left( e_{i} \otimes e_{j} )$$. Can dot products be formed something like $$( e_i \otimes e_j ) \cdot ( e_k \otimes e_l )$$ ?

I've seen some references that say that much as a vector has a (single) direction, a rank two tensors has two directions. Is this always true?

Thanks in Advance
Scott Smith

 

[arkajad]

Yes, you can make dot products.

Two directions are ok for antisymmetric rank 2 tensors - but exactly speaking only "pure" ones. One day you will learn about "wedge product" - it is important.

 

[xGAME-OVERx]

Thanks for the reply! I think I am aware of the wedge product, $${\bf{a}} \wedge {{\bf{b}} = - {\bf{b}} \wedge {\bf{a}}$$, similar to the vector cross product?

So presumably $$( e_i \otimes e_j ) \cdot ( e_k \otimes e_l ) = \delta_{ij} \delta_{kl}$$ for an orthonormal basis?

It is possible to use a basis that isn't orthonormal, say $$u_i$$, and define the dot product as $$( u_i \otimes u_j ) \cdot ( u_k \otimes u_l ) = |u_i| |u_j| \cos \theta_{ij} |u_k| |u_l| \cos \theta_{kl}$$ ?

 

[MathematicalPhysicist]

I think you got mixed with the indices, it should be:


$$( u_i \otimes u_j ) \cdot ( u_k \otimes u_l ) = |u_i| |u_k| \cos \theta_{ik} |u_j| |u_l| \cos \theta_{jl}$$
As in $$( u_i \otimes u_j ) \cdot ( u_k \otimes u_l ) = <u_i , u_k> <u_j, u_l>$$
Or so I remeber it that way.

 

[xGAME-OVERx]

Oops, sorry! Got the j and l indices backwards. Thanks!