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Fix $p\in X$. A linear change of coordinates first identifies ${\omega }_p$ with the standard symplectic form ${\omega }_0$. After shrinking to a star-shaped neighborhood, everyis nondegenerate. The Poincaré lemma gives $\omega -{\omega }_0=d\sigma$, with $\sigma (p)=0$. Define $X_t$ by ${\iota }_{X_t}{\omega }_t=-\sigma$ and let ${\varphi }_t$ be its local flow. Cartan's magic formula givesThus ${\varphi }_1^{\ast }\omega ={\omega }_0$, proving the Darboux theorem.

For a smooth function $H$ on a closed symplectic manifold, its Hamiltonian vector field is defined by ${\iota }_{X_H}\omega =-dH$. The same formula gives ${\mathcal{L}}_{X_H}\omega =-d^{2}H=0$, so the Hamiltonian flow preserves the symplectic form. To move one point to another in a connected $X$, join them by a path, cover the path by finitely many Darboux charts, and in each chart use a cutoff linear Hamiltonian to perform a small translation. Composing these compactly supported Hamiltonian diffeomorphisms proves that symplectomorphisms act transitively on each connected component.

In ${\mathbb{R}}^2$, every embedded curve is a Lagrangian submanifold. Let $L_1,L_2$ be circles enclosing different Euclidean areas. A plane symplectomorphism preserves area and carries the bounded complementary component of one circle to that of its image, so no symplectomorphism maps $L_1$ to $L_2$.

The same phenomenon exists in every ${\mathbb{R}}^{2n}$. Withtake the product toriThe Liouville class of a Lagrangian submanifold has respective period vectors $\pi (1,\dots ,1)$ and $2\pi (1,\dots ,1)$ on these tori. Every symplectomorphism of ${\mathbb{R}}^{2n}$ preserves the Liouville class up to the induced integral change of basis on $H_1(T^n)$, because its pullback changes $\lambda$ only by an exact form. An integral automorphism sends a primitive vector to a primitive vector and therefore cannot send the first period vector to the second. Hence

Part (c) givesThe half-space $x_1>0$ is a simply connected domain, so the Poincaré lemma makes this curl-free field a conservative vector field: it is the gradient of a single-valued scalar potential.

Using $\tan u+\cot u=2/\sin (2u)$,The spherical-coordinate formula from part (b) shows that a potential must satisfy $f_{\varphi }=2$ and may be independent of $r,\theta$. On $x_1>0$ the azimuthal angle has the single-valued branchHence one solution, up to an additive constant, is

If $\nabla \times \mathbf{F}=0$ on a simply connected region containing the bounded trajectory, then the Poincaré lemma gives a scalar potential $U$ with $\mathbf{F}=-\nabla U$. The work iswhich remains bounded when $\mathbf{x}$ and $\mathbf{F}$ remain bounded. Since $H$ is also bounded, $\Delta Q=W-\Delta H$ cannot grow linearly with the observation time.

Thus sustained linear heat dissipation in this setting requires a nonconservative force, and, under the stated simply connectedness assumption,Such a force can perform nonzero work on repeated bounded cycles. On a multiply connected domain, a curl-free force can have nonzero circulation, so the topological assumption is essential.

Integrating by parts makes the $Z$ dependence $-i\int Z{\partial }_{\alpha }{p}^{\alpha }$. Its functional integral imposes the constraint ${\partial }_{\alpha }{p}^{\alpha }=0$. On the simply connected plane, the Poincaré lemma therefore permits the local and global parametrizationSubstitution into $S_2$ givesThis is the Buscher procedure for the translation isometry in $z$, and its T-duality replaces $G_{zz}=V$ by ${\tilde{G}}_{\tilde{z}\tilde{z}}=V^{-1}$.

The de Rham cohomology isThe Poincaré lemma says that every closed positive-degree differential form is locally exact, and is exact on every star-shaped open subset of Euclidean space.

Let $n>1$ and let $\alpha$ be a closed one-form on $M$. The hypothesis gives $\alpha =dg$ on $M\setminus B$. Choose a slightly larger coordinate ball $B^{\prime }$ around $B$. The Poincare lemma gives $\alpha =dh$ on $B^{\prime }$. The annulus $B^{\prime }\setminus B$ is connected when $n>1$, so $d(g-h)=0$ there and $g-h$ is constant. Adjusting $h$ by this constant makes $g$ and $h$ agree on the overlap, and they glue to a global primitive of $\alpha$. Hence $H^1(M)=0$. For $n=1$ the claim fails: remove a closed proper interval from $S^1$. Its complement is an interval and has vanishing first de Rham cohomology, whereas $H^1(S^1)\cong \mathbb{R}$.

Finally choose a nowhere-vanishing $n$-form $\omega$ on $S^n$ and a coordinate $t$ on the finite interval $I$. Every $(n+1)$-form on $S^n\times I$ is $a(x,t)\omega \wedge dt$. Fix $t_0\in I$ and putSince $d_{S^n}A\wedge \omega =0$ for dimensional reasons,Every top-degree form is exact, so $H^{n+1}(S^n\times I)=0$ without using the de Rham theorem.

There is a global-domain issue in the printed question. Write $D=a^2{x}^2+b^2{y}^2>0$ away from the origin. Direct partial differentiation givesThus the differential one-form is a closed differential form for every real $a,b\ne 0$. The Poincaré lemma makes it locally an exact differential form on small neighborhoods avoiding the origin. There is no additional condition such as $a^2=b^2$.

However, the stated domain is the whole punctured plane, which is not simply connected. To test global exactness, use the closed ellipse $x=\cos t/a$, $y=\sin t/b$, $0\le t\le 2\pi$. Here $D=1$ andThe line integral of a globally exact differential form around any closed curve would be zero. ConsequentlyThe distinction is precisely that of the elliptically scaled angular one-form. The remaining parts can be answered consistently using a local angle branch, as follows.

Cover the sphere by complements of its two poles. Each open set is contractible, and their intersection deformation retracts to the equatorial sphere. The Poincaré lemma and homotopy invariance of de Rham cohomology make the two neighboring groups in the Mayer--Vietoris sequence for de Rham cohomology zero, so its connecting map is an isomorphism. Its inverse takes a top-degree form to the difference of its local primitives on the equator, up to the connecting-map sign convention.

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity $u$ has hypersurface orthogonality, then $u^{♭}\wedge d{u}^{♭}=0$ and ${\iota }_{u}d{u}^{♭}=0$. The interior product of the first identity gives $d{u}^{♭}=0$. The Poincaré lemma supplies $u^{♭}=-dt$ locally; spatial coordinate functions constant along the smooth flow of $u$ give $u={\partial }_t$, $g_{tt}=-1$ and $g_{ti}=0$. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.