Let $F=(F_i)_{i=1}^n: X \to Y$ be a map between two manifolds. Suppose that $(U, x_1, \ldots, x_n)$ is a local coordinate on $X$ and $(V, y_1, \ldots, y_m)$ is a local coordinate on $Y$. Suppose that $f=f(y_1, \ldots, y_m)$ is a function on $y_1, \ldots, y_m$. What is the formula for $\frac{\partial}{\partial x_j}(f \circ F)$? Thank you very much.
Is $\frac{\partial}{\partial x_j}(f \circ F) = \sum_{i} \frac{\partial F_i(x)}{\partial x_j} \frac{\partial}{\partial y_i}$? Thank you very much.