So I know basically what a directional derivative is and how to calculate it using the gradient vector, but I'm a bit lost on the more advanced approach of looking at it as a linear transform.
I've read that multivariable calculus is about approximating nonlinear maps with linear ones, and I know that the Jacobian is the matrix associated with the directional derivative. However, I still don't really understand the directional derivative as a linear transform.
For example, what is the input of the transform? Also, what is meant by approximating nonlinear maps by linear maps? I only know a little linear algebra so that might be why I'm so confused.