Rank of map to product manifold is sum of rank of maps?

Question

Suppose $M,N_1, N_2$ are smooth manifolds, $U\subset M$ is open and $F:U\to N_1$ has constant rank $k$, $G:U\to N_2$ has constant rank $l$. Can we then say that $f: U\to N_1\times N_2$, $q\mapsto (F(q),G(q))$ has constant rank $k+l$? I tried writing out the Jacobian of $f$ in local coordinates but the result was a "block" matrix. That is, it looked like the Jacobian of $F$ in the upper left block and the Jacobian of $G$ in the lower right block. Is this correct? Something feels fishy

Answer 1

You probably want another definition of $f$. Seems like you want $f: M \times M \to N_1 \times N_2$. Then your description is correct (the other two blocks are zero) and the rank equation follows from linear independence of rows or columns.