Tensor derivative in continuum mechanics

[enricfemi]

i am currently reading a book about continuum mechanics. they define derivatives of tensor valued functions of tensors in such way:
http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics )
it seems to be related covariant derivative, but i can not figure out its meaning in physics or maths.
i just try to understand why it should be defined in such way.
any talk will be appreciate.

 

[enricfemi]

in other words, if we have a connection, how it acts on tensors.

 

[enricfemi]

ok, it has nothing to do with covariant derivative, seems to be some kinds of direction derivative in vector space. still unware of its meaning.