< Continuum mechanics

Identity 1

Let and be two second order tensors. Show that

Proof:

Using index notation,

Hence,

Identity 2

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 3

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

Proof:

Using index notation,

Hence,

Identity 4

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,


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