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Take $\Omega ={\mathbb{R}}^2$ andIts Jacobian matrix and determinant areThus its total derivative is invertible everywhere, and the inverse function theorem makes it locally invertible. But $F(s,t+2\pi )=F(s,t)$, so it is not injective and has no global inverse on its image. The domain is open and connected as required.

Suppose $D{F}_x$ were invertible at some $x\in \Omega$. The inverse function theorem supplies a neighborhood $V$ of $x$ for which $F(V)$ is a nonempty open subset of ${\mathbb{R}}^2$. But $F(V)\subset M$, and a one-dimensional embedded submanifold of ${\mathbb{R}}^2$ has empty interior. To see this last assertion, a slice chart for an embedded submanifold takes $M$ locally into a coordinate line; a coordinate line cannot contain an open plane neighborhood, and a chart preserves openness. This is a contradiction.

Thus $D{F}_x$ is singular everywhere. Part (c)(i) strengthens singularity to $D{F}_x=0$ everywhere. By (b) and connectedness of $\Omega$,The special form of the Jacobian matrix is essential: an arbitrary differentiable map can have nonconstant image in a curve with derivative of rank one.

The matrix exponential is smooth, and part (a) identifies its derivative at zero as the identity. The inverse function theorem therefore gives a local diffeomorphism from an open neighborhood $U_0$ of zero onto a neighborhood of $I$. Choose a sufficiently small operator norm ball $U\subset U_0$ and let $V=\exp U$. This is again a diffeomorphism onto an open set, since it is a restriction of the original one.

The ball $U$ is preserved by $A\mapsto A^T$ and $A\mapsto -A$. Since $(\exp A{)}^T=\exp (A^T)$ and $(\exp A{)}^{-1}=\exp (-A)$, $V$ is preserved by adjoint and inversion. For its local inverse $L$, injectivity also gives

First establish the limiting law needed by the delta method. For fixed $t$, put $u_n=1/2+t/\sqrt{n}$ and let $N_n$ count how many of the $U_i$ are at most $u_n$. Then $N_n$ has a binomial distribution with expectation $n/2+t\sqrt{n}$ and variance $n/4-t^2$. The event $M_U\le u_n$ is exactly $N_n\ge (n+1)/2$. Its standardized threshold tends to $-2t$, so the central limit theorem for these binomial counts givesHere $\Phi$ is the cumulative distribution function of the standard normal distribution. Equivalently,Use the usual local regularity for the delta method: $f$ is continuous at the population median $\theta$ and $f(\theta )>0$. The inverse function theorem, applied at $F(\theta )=1/2$, givesThereforeThe asymptotic normality gives the requested leading variance approximationHere the variance parameter for the rescaled limiting law is $1/(4f(\theta {)}^2)$, while the source states the corresponding $1/n$ approximation for the unscaled estimator. Strict increase of $F$ alone does not force $f(\theta )>0$; that local condition is needed for this form of the asymptotic distribution of a sample median.

For this change of variables,and $u_{xx}+u_{yy}=0=v_{xx}+v_{yy}$. The second derivative chain rule therefore givesThus the transformation rule for the Laplacian isFor $f=x+y$, its Laplacian is zero. At every point with $(x,y)\ne (0,0)$, division by $4(x^2+y^2)$ provesThis conclusion applies to each local inverse branch of the change of variables: its Jacobian determinant is $4(x^2+y^2)$, so such a branch exists away from the origin by the inverse function theorem. A single globally defined $g$ cannot represent $x+y$ on the entire plane, since $(x,y)$ and $(-x,-y)$ give the same $(u,v)$ but generally opposite values of $x+y$. Nor can a differentiable $g$ represent this $f$ in a neighborhood of the origin: the multivariable chain rule there would give $f_x=f_y=0$, contradicting $f_x=f_y=1$. The requested harmonic function conclusion is therefore a local statement on nonsingular branches.^[ref]

Differentiating the polynomial coordinate map givesIts Jacobian matrix isThe determinant identifies where the inverse function theorem applies.

The determinant of the Jacobian matrix is $s^2+t^2$. Wherever $(s,t)\ne (0,0)$, the inverse function theorem permits a local inverse of the parabolic coordinates transformation, andReading off its entries givesThese are local formulas on a chosen branch: $(s,t)$ and $(-s,-t)$ have the same image, and at the origin the Jacobian matrix is singular, so this inverse-derivative formula does not apply there.

A gauge condition $F(x)=0$ is regular at a representative $x_0$ if its derivative along the gauge orbit is an invertible matrix. The inverse function theorem then makes the gauge-condition values valid local coordinates along that orbit. A unique zero without this regularity is insufficient: the translation action on the line with $F(x)=x^3$ has a unique zero but a zero derivative there. Multiple regular intersections give a Gribov ambiguity rather than a globally unique slice.

A Lagrangian is regular when its velocity Hessian matrix is invertible. The inverse function theorem then makes the fiber derivative $p_i=L_{v^i}$ locally invertible, allowing a local Legendre transform in mechanics to a Hamiltonian. If that fiber derivative is globally invertible, the Lagrangian is called a hyperregular Lagrangian. Regularity alone does not assert global invertibility, and singular Lagrangians may give constrained rather than ordinary unconstrained Hamilton's equations.

A nonempty regular level set $f^{-1}(0)$ for a real smooth function has both positive and negative values immediately beside any of its points, by the inverse function theorem. On its complement the function never vanishes, so the positive and negative regions are disjoint nonempty open sets partitioning that complement. Hence the complement is disconnected. This is a global necessary condition beyond local embeddedness or coorientation.

Series reversion constructs the local inverse of a power series. Substituting an ansatz for the inverse and matching successive powers determines its coefficients recursively.

At a nondegenerate critical point placed at the origin, integral Taylor theorem gives $f(x)-f(0)=x^{T}A(x)x$, where $A(x)$ is a smooth symmetric matrix and $A(0)=\frac{1}{2}\mathrm{Hess}f(0)$. A constant linear change makes $A(0)$ diagonal with entries $\pm 1$. Successive completion of squares, with the Schur complement at each step, produces a smooth invertible triangular matrix $B(x)$ such that $x^{T}A(x)x={\sum }_i{ϵ}_i(B(x)x{)}_i^2$. The pivots keep their nonzero signs near zero. The map $x\mapsto B(x)x$ has invertible derivative at zero, so the inverse function theorem makes it a coordinate system. The signs are the Hessian matrix signature by Sylvester's law of inertia.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

Near a point where a smooth vector field is nonzero, there are coordinates in which it is $\partial /\partial x_1$. Flow a transverse coordinate slice by the local flow; the inverse function theorem makes the resulting map a coordinate chart. This is the first step of the coordinate proof of the Frobenius theorem.

The type-two generating function for a canonical transformation $F_2(q,P)=\cos q\sin P$ defines $p=-\sin q\sin P$ and $Q=\cos q\cos P$. On any branch with $0<q<\pi$ and $\cos P\ne 0$, its inverse in mixed coordinates isThe square root equals $\sin q$ on this branch. Differentiating $F_2$ proves the generating equations, and $F_{2,qP}=-\sin q\cos P\ne 0$ proves local invertibility by the inverse function theorem. Also $d(F_2-PQ)=pdq-PdQ$, so the symplectic form obeys $dq\wedge dp=dQ\wedge dP$. This proves the transformation is canonical locally. It is not a single global chart across the inverse-trigonometric branch boundaries.

Since $F_2$ is independent of time, the transformed Hamiltonian is obtained by substitution, with zero generating-function time correction. An added function of time alone is an irrelevant Hamiltonian freedom, since it leaves Hamilton's equations unchanged.

A smooth embedded submanifold has a neighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle, with the zero section identified with the submanifold. One may prescribe the normal-bundle splitting of the derivative along that section. Choose a Riemannian metric making the splitting orthogonal and use its normal exponential map: its derivative there is the required isomorphism, so the inverse function theorem supplies local diffeomorphisms. A sufficiently small, possibly varying, fibre radius makes these fit into a tubular neighborhood.