Contents 

Initial orthonormal basis:
Deformed orthonormal basis:
We assume that these coincide.
Effect of :
Dyadic notation:
Index notation:
The determinant of the deformation gradient is usually denoted by J and is a measure of the change in volume, i.e.,
Forward Map:
Forward deformation gradient:
Dyadic notation:
Effect of deformation gradient:
Push Forward operation:
Inverse map:
Inverse deformation gradient:
Dyadic notation:
Effect of inverse deformation gradient:
Pull Back operation:

Motion:
Deformation Gradient:
Inverse Deformation Gradient:
Push Forward:
Pull Back:
Recall:
Therefore,
Using index notation:
Right CauchyGreen tensor:
Recall:
Therefore,
Using index notation:
Left CauchyGreen (Finger) tensor:
Green strain tensor:
Index notation:
Almansi strain tensor:
Index notation:
Recall:
Now,
Therefore,
Push Forward:
Pull Back:
We often need to compute the derivative of with respect the the deformation gradient . From tensor calculus we have, for any second order tensor
Therefore,

The derivative of J with respect to the right CauchyGreen deformation tensor () is also often encountered in continuum mechanics.
To calculate the derivative of with respect to , we recall that (for any second order tensor )
Also,
From the symmetry of we have
Therefore, involving the arbitrariness of , we have
Hence,
Also recall that
Therefore,

In index notation,

Another result that is often useful is that for the derivative of the inverse of the right CauchyGreen tensor ().
Recall that, for a second order tensor ,
In index notation
or,
Using this formula and noting that since is a symmetric second order tensor, the derivative of its inverse is a symmetric fourth order tensor we have

