I'm having trouble simplifying the following expression.
$$\frac{\partial ( \hat{e} \cdot A \hat{e} )}{\partial A}$$
Where $A$ is a second order tensor and $\hat{e}$ is a fixed unit vector with no dependence on $A$. It seems like there should be a property that can be used to simplify this. My first approach was to try using product rule on the dot product but realized that doesn't make sense because we're taking a derivative with respect to a tensor.
Write the function in terms of the double-dot product, find the differential, then the gradient $$\eqalign{ \phi &= e\cdot A\cdot e = ee^T:A \cr d\phi &= ee^T:dA \cr \frac{\partial\phi}{\partial A} &= ee^T \cr }$$ The result is more clearly expressed in index notation $$\eqalign{ \frac{\partial\phi}{\partial A_{ij}} &= e_{i}e_{j} \cr }$$