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Under the given rank hypothesis, the necessary and sufficient local condition isVanishing only at the single point would not suffice: an extension preserves the coordinate Poisson brackets throughout its domain. Necessity follows from the coordinate criterion just proved.

For sufficiency, put $Q_i=a_i$ and $Y_i=X_{Q_i}$. Because the symplectic form is a nondegenerate bilinear form at every point, the independent differentials $d{Q}_i$ give independent fields $Y_1,\dots ,Y_n$. They commute, sinceMoreover $d{Q}_j(Y_i)=\{Q_j,Q_i\}=0$, so they span the tangent spaces of the $n$-dimensional fibres of the submersion $Q$. Choose a local section $s(Q)$ transverse to those fibres. Its joint local flow defines coordinates $(Q,t)$ byThe inverse function theorem makes $\Psi$ a local diffeomorphism, and commuting flows give ${\Psi }_{\ast }{\partial }_{t_i}=Y_i$.

In these coordinates ${\iota }_{{\partial }_{t_i}}{\Psi }^{\ast }{\omega }_0=d{Q}_i$. Each flow preserves the symplectic form, by Cartan's magic formula. Thus its coefficients are independent of $t$, andwhere $\beta =s^{\ast }{\omega }_0$ is a closed differential two-form on the base. On a small ball, the Poincare lemma gives $\beta =d\alpha$, with $\alpha ={\sum }_i{\alpha }_i(Q)d{Q}_i$. DefineThenConsequently the original coordinates are extended to the required canonical transformation $(Q,P)$, with the prescribed first half $Q=a$. Its Jacobian matrix is invertible by construction. This is the canonical completion of independent commuting functions. For the printed $C^2$ data, the flows and resulting canonical transformation can be taken $C^1$, which is sufficient to preserve the symplectic form; smooth data give smooth coordinates.

Use $(y,w)$ to label the initial point in phase space. The characteristic equations for a transport equation are $\dot{X}=a(V)$ and $\dot{V}=0$, with $X(s)=y$ and $V(s)=w$. Since the velocity remains fixed, the ordinary differential equations integrate directly:This characteristic flow map exists for all real times, even if $a$ grows rapidly at infinity: each trajectory samples just one fixed velocity. Its inverse is $S_{t,s}$, and its Jacobian matrix is the block triangular matrixConsequently its Jacobian determinant is one. The nonzero Jacobian determinant of $a$ is unnecessary for this construction.

Fix the standard Jacobian matrix convention $(Da{)}_{ij}={\partial }_{v_j}{a}_i$. With column gradients, the commuting velocity derivative for kinetic transport isIf the symbol $J_a$ is instead defined with derivative indices as rows, the printed expression is this same formula. Under the standard convention it needs a transpose.

To prove the claim, put $D={\partial }_t+a(v)\cdot {\nabla }_x$. Differentiating $Df=0$ in $v_j$ givesThe terms cancel, so each component of $G$ satisfies the transport equation. Multiplication by the smooth coefficient $Da$ and differentiation preserve the same uniform compact support, hence $G\in {\mathcal{C}}_T$ componentwise. At $t=0$, $G(0)={\nabla }_v{f}_{\mathrm{in}}$. The L2 norm calculation in part (d), summed over components, givesOne can also differentiate the explicit solution: the $t(Da{)}^T{\nabla }_{x}f$ term cancels the derivative of $x-ta(v)$, leaving the initial velocity derivative transported along the characteristic curves.

The transpose matters. For the shear $a(v)=(v_1+v_2,v_2)$, $\det Da=1$ but $Da\ne (Da{)}^T$. The printed field satisfies $Dg=(Da-(Da{)}^T){\nabla }_{x}f$, which is nonzero for a generic compactly supported datum. The untransposed formula is false with the standard convention.

Differentiating the explicit characteristic flow map gives the block triangular Jacobian matrixIts Jacobian determinant is the product of its diagonal block determinants. Therefore

The phase plane system is $\dot{x}=y$, $\dot{y}=-cy-\sin x$. Its equilibrium points are $(k\pi ,0)$ for every integer $k$. At such a point the Jacobian matrix and characteristic equation areFor even $k$, the equilibrium point is a stable focus for $0<c<2$, with eigenvalues $-c/2\pm i\sqrt{1-c^2/4}$; it is a stable node for $c>2$, with two distinct negative real eigenvalues. The excluded value $c=2$ gives the repeated critical case.

For odd $k$, the eigenvalues are $(-c\pm \sqrt{c^2+4})/2$. One is positive and one negative, so each is a saddle equilibrium for every $c>0$. These classifications follow from the linearization of a dynamical system, since all the relevant eigenvalues have nonzero real part for $c>0$. The even equilibria remain hyperbolic at $c=2$ too, although their linearization then has a repeated eigenvalue; that critical case is excluded from the requested classification.

Work in a translation-invariant pure thermodynamic phase, and write ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ to distinguish inverse temperature from the order-parameter exponent. The connected correlation function isAway from criticality in the massive scalar order channel, it decays exponentially, possibly multiplied by an algebraic factor. An exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when that limit exists. At a continuous transition $\xi$ diverges and the critical decay is algebraic. In an ordered symmetric mixture, a nondecaying contribution can remain; selecting a pure branch prevents confusing it with connected critical fluctuations.

Let $M={\sum }_{\mathbf{n}}{\sigma }_{\mathbf{n}}$ and $m=⟨M⟩/N$. Differentiating the partition function with the energy term $-hM$ gives the correlation-function susceptibility sum ruleIf the source is instead the dimensionless ${\beta }_{\mathrm{th}}h$, the explicit inverse-temperature factor is absent. In continuum physical coordinates the lattice sum is approximately $a^{-D}\int d^{D}rG(\mathbf{r})$.

Choose a normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ such that ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$, replacing the sum by an integral for continuous variables. A deterministic example is a product of delta functions imposing that each coarse variable equals the average of spins in a block of $b^D$ sites, with its field normalization included consistently. Define the effective Boltzmann weight byKernel normalization makes the blocked partition function exactly equal to the original. Its coarse-grained variables retain the long-distance observables through their defining relation to the original fields. An exact RG step generally generates all symmetry-allowed operators; keeping only a small coupling set is an approximation, not an exact closure assumption. The geometrical coarse lattice hasso its physical volume is unchanged. A subsequent coordinate rescaling may restore the numerical lattice spacing to its initial value, which is the equivalent rescaled-coordinate convention.

Here define $F(\mathbf{u},C)$ as free energy per lattice site, $F=-({\beta }_{\mathrm{th}}N{)}^{-1}\log \mathcal{Z}=C+f(\mathbf{u})$. Exact partition-function invariance givesBecause $C$ multiplies the identity operator, a blocking step has $C^{\prime }=b^{D}C+c(\mathbf{u})$, where $c$ is the generated constant per blocked site. With $g(\mathbf{u})=b^{-D}c(\mathbf{u})$ this impliesThe identity-operator contribution to renormalization-group free energy includes the eliminated modes' entropy, connected vacuum terms and field-measure normalization. It cannot be discarded when calculating an absolute free energy, even though it cancels out of normalized correlation functions.

There is a useful convention behind the word “singular” in this inhomogeneous equation. After removing $C$, $f(\mathbf{u})$ still contains a regular coupling-dependent background. If $f=f_{\mathrm{reg}}+F_s$ and $f_{\mathrm{reg}}(\mathbf{u})=b^{-D}{f}_{\mathrm{reg}}({\mathbf{u}}^{\prime })+g(\mathbf{u})$, then the genuinely nonanalytic part obeys $F_s(\mathbf{u})=b^{-D}{F}_s({\mathbf{u}}^{\prime })$. Thus the printed inhomogeneous representative and homogeneous singular scaling below are consistent after background subtraction. At resonances or marginal points this subtraction may leave additive or multiplicative logarithms; a strictly homogeneous pure-power form is then qualified accordingly.

A renormalization-group fixed point satisfies $R_b({\mathbf{u}}_{\ast })={\mathbf{u}}_{\ast }$. Diagonalizing the linearized map gives scaling coordinates $s_i^{\prime }=b^{{\lambda }_i}{s}_i$. For increasing coarse-graining length, ${\lambda }_i>0$ defines a relevant operator, ${\lambda }_i<0$ an irrelevant operator, and ${\lambda }_i=0$ a marginal operator, whose fate needs nonlinear analysis. Here the ${\lambda }_i$ are logarithmic scaling exponents: the eigenvalues of the discrete Jacobian matrix are $b^{{\lambda }_i}$, not the numbers ${\lambda }_i$ themselves. The critical surface is the stable manifold flowing into the critical fixed point after every relevant scaling field is tuned to zero. A repulsive renormalization-group trajectory leaves the fixed point along relevant directions and flows toward a different long-distance phase.

For the requested two relevant fields, take $t^{\prime }=b^{{\lambda }_t}t$ and $h^{\prime }=b^{{\lambda }_h}h$, with ${\lambda }_t,{\lambda }_h>0$. Ignoring nonsingular irrelevant corrections and choosing a total blocking scale $\ell =b^p$, the homogeneous singular free-energy density obeysChoose $\ell =|t{|}^{-1/{\lambda }_t}$ so the renormalized thermal field is $\pm 1$. This provesMetric factors and regular analytic redefinitions of the scaling fields only change amplitudes. Likewise $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$, so at $h=0$ the correlation-length critical exponent is $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s$ give the heat-capacity critical exponent $\alpha =2-D/{\lambda }_t$, proving the hyperscaling relationTwo field derivatives also give $\gamma =(2{\lambda }_h-D)/{\lambda }_t$; one field derivative gives $\beta =(D-{\lambda }_h)/{\lambda }_t$, and using $h$ to set the blocking scale on the critical isotherm gives $\delta ={\lambda }_h/(D-{\lambda }_h)$. These statements assume that no dangerously irrelevant coupling makes the scaling function singular as it is removed. In particular the naive hyperscaling form need not describe the quartic ordered phase above four dimensions, where its stabilizing quartic coupling is dangerously irrelevant. That restriction reconciles this derivation with the mean-field exponents and Question 3.

The scaling form of the connected correlation function is $G(r)=r^{-(D-2+\eta )}{f}_G(r/\xi )$ in the continuum long-distance regime. For $r\ll \xi$, $f_G$ tends to a finite nonzero constant. For $r\gg \xi$, it decays exponentially with a possible algebraic prefactor; an isolated massive pole gives the familiar Ornstein--Zernike correlation function tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ with a temperature-dependent amplitude. Exponential decay is the essential feature; the same critical algebraic power need not remain the exact large-distance prefactor.

Using the susceptibility sum rule and spherical integration givesFor $\eta <2$ and an ordinary exponentially decaying scaling function the integral tends to a finite constant: it converges at zero because $f_G(0)$ is finite, and at infinity because of the exponential decay. Microscopic distances add a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, establishing the Fisher scaling relation

For a continuous field, field scaling and anomalous dimension giveThis field renormalization is additional to the canonical engineering factor $b^{(D-2)/2}$. It makes $G(r)=b^{-2x_{\sigma }}{G}^{\prime }(r/b)$. Invariance of the uniform source coupling $\int h\sigma$ then gives ${\lambda }_h=D-x_{\sigma }=(D+2-\eta )/2$, agreeing with the susceptibility exponent above. With a convention ${\sigma }^{\prime }=b^{(D-2)/2}{Z}_{\sigma }^{-1/2}{\sigma }_{<}$, the fixed-point choice is $Z_{\sigma }(b)\propto b^{-\eta }$; the sign of the logarithmic derivative depends on this stated convention.

To find the threefold phase-locked equilibria, write $C=R{e}^{i\theta }$ with $R>0$; separating real and imaginary parts givesA steady state satisfiessoWriting $y=R_0^2$ and $D=1+4\mu -4{\omega }^2$, the amplitude branches areThere are two distinct positive roots for $\mu >{\omega }^2-1/4$, except at $(\mu ,\omega )=(0,0)$, where $y_{-}=0$ and only the upper root is nonzero. Indeed their sum is positive in this range and their product is ${\mu }^2+{\omega }^2$. Below the fold condition there are none. At equality, there is one positive repeated root $y={\omega }^2+1/4$, the saddle-node bifurcation limit.

For each positive amplitude, the sine and cosine determine $3{\theta }_0$ modulo $2\pi$, giving three phases separated by $2\pi /3$. Thus the source's “two states” means two amplitude branches modulo the threefold spatial symmetry. Generically there are six nonzero complex equilibria, three on each branch, not literally two.

The polar Jacobian matrix at an equilibrium isOn the lower branch, $\det J=-3y_{-}\sqrt{D}<0$, so it is a saddle equilibrium and unstable. On the upper branch, $\det J=3y_{+}\sqrt{D}>0$ andbecause existence implies $\mu >-1/4$. Therefore every upper-branch equilibrium is asymptotically stable, and every lower-branch equilibrium is a saddle equilibrium, away from the degenerate endpoints. The eigenvalues are unchanged by the smooth polar coordinate transformation at $R_0>0$. The origin, not covered by those coordinates, has eigenvalues $\mu \pm i\omega$ and is stable for $\mu <0$ and unstable for $\mu >0$.

On an affine chart $\mathrm{Spec}A\subseteq V$, the Module of Kähler differentials ${\Omega }_{A/k}$ is generated by symbols $da$ subject to $k$-linearity and $d(ab)=adb+bda$. It represents $k$-derivations. These modules commute with localization, so their associated quasi-coherent sheaves glue to the Kähler differential sheaf ${\Omega }_{V/k}^1$. It is coherent: if $A=k[x_1,\dots ,x_N]/I$ and $I=(f_1,\dots ,f_s)$, then it has the finite presentation of a moduleFor a closed point $P$, put $R={\mathcal{O}}_{V,P}$ with maximal ideal $\mathfrak{m}$ and residue field $k$. The Zariski tangent space is ${\mathrm{Hom}}_k(\mathfrak{m}/{\mathfrak{m}}^2,k)$, equivalently ${\mathrm{Der}}_k(R,k)$. A derivation kills constants and ${\mathfrak{m}}^2$, so it factors through $a\mapsto a-a(P)$; conversely every linear functional on $\mathfrak{m}/{\mathfrak{m}}^2$ defines such a derivation by the product rule. The universal property of Kähler differentials therefore identifies this space withIn particular, ${\Omega }_{V,P}^1/{\mathfrak{m}}_P{\Omega }_{V,P}^1\cong {\mathfrak{m}}_P/{\mathfrak{m}}_P^2$, the algebraic cotangent space.

A point is a smooth point of a variety when its local ring is a regular local ring; over this algebraically closed field this says ${\dim }_k{T}_{V,P}=n=\dim V$. Tensor the finite presentation of a module above with $k(P)$. The tangent space is the kernel of the Jacobian matrix $J(P)$, hence has dimension $N-\mathrm{rank}J(P)$. This proves the Jacobian criterionFor a smooth irreducible variety $V$, choose at each point an invertible $(N-n)$-minor and shrink the affine chart so that it remains invertible. Its relations eliminate $N-n$ of the generators of ${\Omega }_{A/k}$, yielding a surjection $A^n\to {\Omega }_{A/k}$. At the generic point, its target has dimension $n$: over a perfect ground field, a separating transcendence basis of the function field has differentials forming a basis. Equivalently this follows from the assumed density of the smooth locus. The kernel therefore becomes zero over the fraction field of the integral domain $A$. As a submodule of $A^n$, it is a torsion-free module, so it is already zero. Thus these maps give local isomorphisms with ${\mathcal{O}}_V^n$, proving local freeness of differentials on a smooth variety with rank $n$.

For an affine chart $\mathrm{Spec}A\subseteq V$, write $W\cap \mathrm{Spec}A=\mathrm{Spec}(A/I)$ and $\mathcal{M}=\tilde{M}$. The restriction of a module sheaf to a closed subvariety is $\widetilde{M{\otimes }_{A}A/I}=\widetilde{M/IM}$. The Conormal exact sequence for Kähler differentials isIt follows from the generators and relations: passing to $A/I$ imposes precisely the additional relations $df=0$ for $f\in I$. The first map is well defined because $d(I^2)$ lies in $I{\Omega }_{A/k}$. Glue these exact module sequences, using exactness of localization, to obtainIn the final assertion, interpret locally principal subvariety as a proper local hypersurface. Its ideal on each chart is $(f)$ with $f\ne 0$. Since $V$ is an irreducible variety and reduced, $f$ is a non-zero-divisor, and $A/(f)\to (f)/(f^2)$, $\bar{a}\mapsto af$, is an isomorphism. These local rank-one descriptions make the conormal sheaf ${\mathcal{I}}_W/{\mathcal{I}}_W^2$ invertible.

Because $W$ is not contained in the singular locus of $V$, there is a dense open subset of $W$ where both $V$ and $W$ are smooth varieties. At a closed point there, $\dim W=\dim V-1$ by the Krull principal ideal theorem. The tangent description $T_{W,P}=\ker (df:T_{V,P}\to k)$ then forces $df\ne 0$. Hence the conormal map is injective at the generic point of $W$. Its kernel is a subsheaf of a line bundle on the integral variety $W$, so it is a torsion-free sheaf; a torsion-free sheaf with zero generic fibre is zero. This proves conormal injectivity for a generically smooth Cartier divisor. If zero equations were allowed in the phrase locally principal, $W=V$ would be a counterexample to invertibility; the proper-hypersurface convention is essential.