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The local nonsingular codimension-one Pfaff–Frobenius theorem

Statement

Let $U \subseteq R^{n + 1}$ , $n \geq 1$ , be open and let $ω = \sum_{i = 1}^{n} a_{i} d x_{i} + b d z$ be a $C^{2}$ one-form with $ω_{q_{0}} \neq 0$ at $q_{0} \in U$ . Relabel coordinates so one coefficient $b$ is nonzero at $q_{0}$ and shrink to a neighborhood where it stays nonzero. The following conditions, both local at $q_{0}$ , are equivalent: (i) there is a neighborhood $U_{0}$ of $q_{0}$ on which $ω \land d ω = 0$ ; (ii) there is a neighborhood $W$ of $q_{0}$ and $μ \in C^{1} (W)$ nowhere zero and $S \in C^{2} (W)$ with $d S = μ ω$ on $W$ . The theorem gives no global exactness or thermodynamic calibration.

Facts & Assumptions

Given: The open Euclidean domain, the coefficient regularity, and nonsingularity near $q_{0}$ .

[F1]

A nonsingular Pfaff one-form and its horizontal paths: $ω_{q}$ is a covector and $H_{q} = \ker ω_{q}$ is its rank- $n$ horizontal subspace; $d ω$ and the alternating products are the stated coordinate forms.

[F2]

Local flows by a contraction argument: a $C^{1}$ vector field has a unique local flow with local inverse and Lipschitz dependence.

[F3]

Variational equations and commuting local flows: for $C^{2}$ fields, local flows are $C^{2}$ and commute when their bracket vanishes.

[F4]

An exact nonzero multiple satisfies the Pfaff condition: $d S = μ ω$ with $S \in C^{2}$ and nowhere-zero $μ \in C^{1}$ implies $ω \land d ω = 0$ .

[F5]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composition is the composition of derivatives.

[L1]

Local graph form at a nonsingular derivative minor (the current carrier is draft, inventoried on the batch-4 A page, order 268): for $1 \leq q \leq m$ , $F : O \subseteq R^{m} \to R^{q}$ is $C^{1}$ , $F (x_{*}) = 0$ , and some $q \times q$ minor of $D F (x_{*})$ is invertible; if $q < m$ , after permuting $x = (u, v) \in R^{m - q} \times R^{q}$ , there are open neighborhoods and a unique $C^{1}$ graph $v = φ (u)$ equivalent to $F (u, v) = 0$ . Its inspected proof reduces to the published Euclidean implicit-function theorem in the $q < m$ case and to the published inverse-function theorem when $q = m$ . In this use $m = 2 (n + 1)$ and $q = n + 1 < m$ , and the solved-coordinate minor is $D Φ (0)$ ; the interface matches. The rank carrier's proof contract and dependencies pass scoped checks, but its phase-number precheck reports a canonical-label repair and its Step1 readiness record is missing, so this cross-batch use remains conditional.

Proof

Given: Translate $q_{0}$ to $(0, z_{0})$ . Restrict all fields and flows to a sufficiently small coordinate neighborhood of $q_{0}$ .

For (i) $\Rightarrow$ (ii), assume (i) on a neighborhood of $q_{0}$ and shrink the working coordinate neighborhood to lie inside it.

1.1givenF1construct

Since $ω_{q_{0}} \neq 0$ , one coordinate coefficient is nonzero at $q_{0}$ ; relabel that coordinate as $z$ and shrink the neighborhood so $b \neq 0$ . Put $c_{i} : = - a_{i} / b$ and $X_{i} : = \partial_{x_{i}} + c_{i} \partial_{z}$ . Then $X_{i} \in C^{2}$ , $ω (X_{i}) = a_{i} + b c_{i} = 0$ , and the $X_{i}$ are a basis of $\ker ω$ because their first $n$ components form the standard basis.

1.2F1algebra

Define $[X, Y] : = D Y X - D X Y$ . Coordinate expansion of the displayed formula for $d ω$ gives $d ω (X, Y) = X [ω (Y)] - Y [ω (X)] - ω ([X, Y])$ . Thus for horizontal fields $d ω (X_{i}, X_{j}) = - ω ([X_{i}, X_{j}])$ . Also $[X_{i}, X_{j}]$ has zero $x$ -components, hence is vertical. Evaluating the three-form on $(\partial_{z}, X_{i}, X_{j})$ gives $(ω \land d ω) (\partial_{z}, X_{i}, X_{j}) = b d ω (X_{i}, X_{j})$ . Under (i), $d ω (X_{i}, X_{j}) = 0$ ; then $ω ([X_{i}, X_{j}]) = 0$ , and a vertical vector $v \partial_{z}$ has $ω (v \partial_{z}) = v b$ , so $[X_{i}, X_{j}] = 0$ .

2.1F2F3step 1.2algebra

By [F2]–[F3] and step 1.2, the fields $X_{i}$ have commuting local $C^{2}$ flows $ϕ_{i}^{t}$ . Define $Φ (x, η) : = ϕ_{n}^{x_{n}} \circ \dots \circ ϕ_{1}^{x_{1}} (0, η)$ for $(x, η)$ near $(0, z_{0})$ . Each $X_{i}$ changes only $x_{i}$ , at rate one, so the first $n$ coordinates of $Φ (x, η)$ are exactly $x$ . Commutation and the transported-field identity in [F3] give $\partial_{x_{i}} Φ = X_{i} (Φ)$ . At $(0, z_{0})$ the columns of $D Φ$ are $X_{1}, \dots, X_{n}, \partial_{z}$ ; their first $n$ components form $I_{n}$ and the last column is $\partial_{z}$ , so $D Φ$ is invertible.

3.1L1F3F5step 2.1algebra

Let $N = n + 1$ . On an open product neighborhood of $(u_{0}, y_{0}) = ((0, z_{0}), (0, z_{0}))$ where $Φ$ is defined, set $H (u, y) : = Φ (u) - y$ . Then $H (u_{0}, y_{0}) = 0$ and its $u$ -derivative $D_{u} H = D Φ (u_{0})$ is invertible. Apply [L1] with ambient dimension $2 N$ and target dimension $N$ , permuting the input coordinates to $(y, u)$ so the $y$ block is free and the solved $u$ block has the invertible minor: conditionally, the zero set is the graph of a unique local $C^{1}$ map $G (y)$ , so $Φ (G (y)) = y$ . Since $Φ$ is $C^{2}$ , differentiating this identity by [F5] gives $D Φ (G (y)) D G (y) = I$ and hence $D G (y) = (D Φ (G (y)))^{- 1}$ . The right side is $C^{1}$ (matrix inverse entries are cofactors divided by the nonzero determinant), so $G$ is $C^{2}$ .

4.1step 2.1step 3.1F3algebra

Let $S (x, z)$ be the last coordinate $η$ of $G (x, z)$ . Then $S \in C^{2}$ , $S (Φ (x, η)) = η$ , and $S_{z} (0, z_{0}) = 1$ ; after shrinking, $S_{z} \neq 0$ . Differentiating along $X_{i}$ gives $S_{x_{i}} + c_{i} S_{z} = 0$ , so $d S = S_{z} (d z - \sum_{i} c_{i} d x_{i}) = (S_{z} / b) ω$ . Thus $μ : = S_{z} / b$ is $C^{1}$ and nowhere zero. This proves (ii), conditionally on [L1].

5.1F4∎

Conversely, if (ii) holds, [F4] gives $ω \land d ω = 0$ on that neighborhood, proving (i).

Remarks

The proof builds a local potential by commuting flows, rather than inferring a global entropy. A nonzero period on a punctured domain can obstruct a global potential even when a local integrating factor exists. If a physical theory later identifies $S$ with entropy, its sign, units, and physical meaning require separate hypotheses and calibration.

The local nonsingular codimension-one Pfaff–Frobenius theorem

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources