Continuum mechanics/Tensor algebra identities

Identity 1Edit

Let   and   be two second order tensors. Show that

 

Proof:

Using index notation,

 

Hence,

 

Identity 2Edit

Let   be a second order tensor and let   and   be two vectors. Show that

 

Proof:

It is convenient to use index notation for this. We have

 

Hence,

 

Identity 3Edit

Let   and   be two second order tensors and let   and   be two vectors. Show that

 

Proof:

Using index notation,

 

Hence,

 

Identity 4Edit

Let   be a second order tensors and let   and   be two vectors. Show that

 

Proof:

For the first identity, using index notation, we have

 

Hence,

 

For the second identity, we have

 

Therefore,

 

Now,   and  . Hence,

 

Therefore,