I have a function $\phi(r,t)=p(r) \cdot k(t) $
What would be $\frac{\partial \phi}{\partial r} $ ?
Can I rewrite it this way: $ \frac{\partial \phi}{\partial r}=\frac{\partial(p(r) \cdot k(t))}{\partial r}=\frac{\partial p(r)}{\partial r} \cdot k(t) $ ?
This is almost true. Since you are in a chart other terms might appear because a scalar product is defined with a metric $g_{ij}p^i k^j$. In $\mathbb{R}^3$ this amounts to
\begin{equation} \frac{\partial}{\partial r}(A(r)\cdot B) = B \cdot \frac{\partial A(r)}{\partial r} + B\times \left(\frac{\partial}{\partial r}\times A(r)\right) \end{equation}