Local Normal Form
If ƒ: M → N is a submersion at p and ƒ(p) = q ∈ N then there exist an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1,…,xm) at p and (x1,…,xn) at q such that ƒ(U) = V and the map ƒ in these local coordinates is the standard projection
It follows that the full pre-image ƒ−1(q) in M of a regular value q ∈ N under a differentiable map ƒ: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q ∈ N if the map ƒ is a submersion.
Read more about this topic: Submersion (mathematics)
Famous quotes containing the words local, normal and/or form:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“Our normal waking consciousness, rational consciousness as we call it, is but one special type of consciousness, whilst all about it, parted from it by the filmiest of screens, there lie potential forms of consciousness entirely different.”
—William James (1842–1910)
“To do the opposite of something is also a form of imitation, namely an imitation of its opposite.”
—G.C. (Georg Christoph)