inner product of gradient

Question

I am trying to understand the below equation:

$ \langle \nabla (f \circ R)(x) , R^T h \rangle = \lim_{\epsilon\to 0} \frac{(f \circ R) (x + \epsilon R^T h) - (f \circ R) (x) }{\epsilon}$

where R is a 2x2 rotation matrix, h be a $R^2$ vector, and f $\in$ $C^2(\Omega;R)$ with $\Omega \subset R^2$ be a real valued function.

My question is what is the LHS trying to do? I suppose it is a inner product of the gradient which is similar to projection? And how did we get the equation on the RHS? It looks similar to the definition of gradient.

Answer 1

The LHS is essentially the projection of the gradient of the function $(f \circ R)(x)$ in the direction of the vector $R^T h $ or, in other words, the directional derivative of the function in the direction of this vector. The RHS is the definition of such derivative using the classical incremental quotient in the direction of the vector.