Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.It is essential for V to be simply connected for the function f to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the 'realification' of the complex square function Since the differential at a point (for a differentiable function) Diffeomorphisms are necessarily between manifolds of the same dimension. Imagine f going from dimension n to dimension k. If n < k then Dfx could never be surjective; and if n > k then Dfx could never be injective. In both cases, therefore, Dfx fails to be a bijection.If Dfx is a bijection at x then f is said to be a local diffeomorphism (since, by continuity, Dfy will also be bijective for all y sufficiently close to x).Given a smooth map from dimension n to dimension k, if Df (or, locally, Dfx) is surjective, f is said to be a submersion (or, locally, a 'local submersion'); and if Df (or, locally, Dfx) is injective, f is said to be an immersion (or, locally, a 'local immersion').A differentiable bijection is not necessarily a diffeomorphism. f(x) = x3, for example, is not a diffeomorphism from R to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.When f is a map between differentiable manifolds, a diffeomorphic f is a stronger condition than a homeomorphic f. For a diffeomorphism, f and its inverse need to be differentiable; for a homeomorphism, f and its inverse need only be continuous. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism. In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth. Given two manifolds M and N, a differentiable map f : M → N is called a diffeomorphism if it is a bijection and its inverse f−1 : N → M is differentiable as well. If these functions are r times continuously differentiable, f is called a Cr-diffeomorphism. Two manifolds M and N are diffeomorphic (symbol usually being ≃) if there is a diffeomorphism f from M to N. They are Cr diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Given a subset X of a manifold M and a subset Y of a manifold N, a function f : X → Y is said to be smooth if for all p in X there is a neighborhood U ⊂ M of p and a smooth function g : U → N such that the restrictions agree g | U ∩ X = f | U ∩ X {displaystyle g_{|Ucap X}=f_{|Ucap X}} (note that g is an extension of f). f is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth. If U, V are connected open subsets of Rn such that V is simply connected, a differentiable map f : U → V is a diffeomorphism if it is proper and if the differential Dfx : Rn → Rn is bijective at each point x in U. f : M → N is called a diffeomorphism if, in coordinate charts, it satisfies the definition above. More precisely: Pick any cover of M by compatible coordinate charts and do the same for N. Let φ and ψ be charts on, respectively, M and N, with U and V as, respectively, the images of φ and ψ. The map ψfφ−1 : U → V is then a diffeomorphism as in the definition above, whenever f(φ−1(U)) ⊂ ψ−1(V). Since any manifold can be locally parametrised, we can consider some explicit maps from R2 into R2. In mechanics, a stress-induced transformation is called a deformation and may be described by a diffeomorphism.A diffeomorphism f : U → V between two surfaces U and V has a Jacobian matrix Df that is an invertible matrix. In fact, it is required that for p in U, there is a neighborhood of p in which the Jacobian Df stays non-singular. Since the Jacobian is a 2 × 2 real matrix, Df can be read as one of three types of complex number: ordinary complex, split complex number, or dual number. Suppose that in a chart of the surface, f ( x , y ) = ( u , v ) . {displaystyle f(x,y)=(u,v).} The total differential of u is