I am having a problem understand a particular result in tensor calculus. It is quite a simple result. I shall set the scene.
It is possible to define a double contraction of two second order tensors , which is given by
Now we come onto the idea of taking the gradient of a scalar-valued function of a second order tensor. Consider to be a scalar-valued function of second order tensors, . It is a non-linear function, and we aim to approximate at by a linear function. Taylor expanding at apparently yields
where is given by the double contraction:
Now I have tried and tried to understand how this follows from some form of Taylor expansion, but I'm not getting anywhere. Can someone point me in the right direction?