I am reading the unit on manifolds in Flemings Functions of Several Variables. Here's a snippet from the page:
I am not sure of the last paragraph: how is the dimension of the intersections of the tangent spaces $r + s - n$. Also, shouldn't it be the case that the intersection of the tangent spaces is empty. If $\mathbf{h}$ belongs to the intersection,
$\mathbf{h} . \text{grad} \mathbf{\Psi} = 0$ and $\mathbf{h} . \text{grad} \mathbf{\Theta} = 0$,
and this will make the gradients to be linearly dependent, which shouldn't be the case.