These definitions extend to differential maps between differentiable manifolds in the following way. Let formula_26 be a differential map between two manifolds and of respective dimensions "m" and "n". In the neighborhood of a point of and of , charts are diffeomorphisms formula_27 and formula_28 The point is critical for if formula_29 is critical for formula_30 This definition does not depend on the choice of the charts because the transitions maps being diffeomorphisms, their Jacobian matrices are invertible and multiplying by them does not modify the rank of the Jacobian matrix of formula_30 If "M" is a Hilbert manifold (not necessarily finite dimensional) and "f" is a real-valued function then we say that "p" is a critical point of "f" if "f" is "not" a submersion at "p".