Continuum mechanics/Tensor-vector identities

Tensor-vector identity - 1Edit

 

Proof:

Using the identity   we have

 

Also, using the definition   we have

 

Therefore,

 

Using the identity   we have

 

Finally, using the relation  , we get

 

Hence,

 

Tensor-vector identity 2Edit

Let   be a vector field and let   be a second-order tensor field. Let   and   be two arbitrary vectors. Show that

 

Proof:

Using the identity   we have

 

From the identity  , we have  .

Since   is constant,  , and we have

 

From the relation   we have

 

Using the relation  , we get

 

Therefore, the final form of the first term is

 

For the second term, from the identity   we get,  .

Since   is constant,  , and we have

 

From the definition  , we get

 

Therefore, the final form of the second term is

 

Adding the two terms, we get

 

Therefore,