Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-23 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Kinetic/internal/total energy and boundary work

Physical scope

Smooth positive-density balance branch: total energy splits into kinetic and internal energy; conservative potential and compatible no-slip/insulating or periodic data give a conditional global conservation law.

Statement

Let $I \subseteq R$ be an open interval containing $0$ , let $B \subseteq R^{3}$ be a nonempty bounded open reference region with $C^{1}$ boundary (locally a $C^{1}$ graph with the region on one side), and let $U \subseteq R^{3}$ be an open neighborhood of $\overline{B}$ . Let $χ \in C^{2} (I \times U; R^{3})$ be a motion with orientation-preserving diffeomorphic time slices (bijections onto open images with $C^{2}$ inverses and positive Jacobian determinant) and $χ (0, a) = a$ . Here $C^{r}$ means all coordinate partial derivatives through order $r$ are continuous. Set $Ω_{t} = χ_{t} (B)$ and $O = {(t, x) : t \in I, x \in χ_{t} (U)}$ . Use SI units in a classical Cartesian inertial frame. Let $ρ_{0} \in C^{1} (\overline{B})$ and $ρ \in C^{1} (O; (0, \infty))$ be positive densities, and let $u : O \to R^{3}$ , $e : O \to R$ , $q : O \to R^{3}$ , and $σ : O \to R^{3 \times 3}$ be $C^{1}$ , with $u (t, χ (t, a)) = \partial_{t} χ (t, a)$ for every $(t, a) \in I \times U$ ; let $b : O \to R^{3}$ and $r : O \to R$ be continuous. Their units are respectively kilograms per cubic metre, metres per second, joules per kilogram, watts per square metre, pascals, metres per second squared, and watts per kilogram. Define $k : = ∣ u ∣^{2} / 2$ , $E : = e + k$ , $D_{x} u = (\partial_{x_{j}} u_{i})_{i j}$ , $D : = (D_{x} u + (D_{x} u)^{T}) / 2$ , $D_{t} : = \partial_{t} + u \cdot \nabla_{x}$ , and $A : B : = \sum_{i, j} A_{i j} B_{i j}$ . Use rowwise divergence $(div σ)_{i} = \sum_{j} \partial_{x_{j}} σ_{i j}$ and vector divergence $div q = \sum_{j} \partial_{x_{j}} q_{j}$ . For every nonempty bounded open $V$ whose closure is compact in $B$ and whose $C^{1}$ boundary is locally a graph with the region on one side, put $P_{t} = χ_{t} (V)$ and let $n$ be its outward Euclidean unit normal. Use Euclidean surface measure $d S$ with chart density $\sqrt{\det (D Y^{T} D Y)}$ .

Adopt, on each such material region,

$\int_{P_{t}} ρ d x = \int_{V} ρ_{0} d a, \frac{d}{d t} \int_{P_{t}} ρ u d x = \int_{P_{t}} ρ b d x + \int_{\partial P_{t}} σ n d S,$

$\frac{d}{d t} \int_{P_{t}} x \times ρ u d x = \int_{P_{t}} x \times ρ b d x + \int_{\partial P_{t}} x \times σ n d S,$

$\frac{d}{d t} \int_{P_{t}} ρ E d x = \int_{P_{t}} (ρ b \cdot u + ρ r) d x + \int_{\partial P_{t}} (u \cdot σ n - q \cdot n) d S .$

The angular-momentum balance is about a fixed inertial origin; assume no surface couples or intrinsic spin. Assume also the Axiom of Choice. On the balanced physical region $O_{B} : = {(t, x) : t \in I, x \in Ω_{t}}$ , angular momentum gives $σ = σ^{T}$ , and the local total-energy balance is

$\partial_{t} (ρ E) + div (ρ E u - σ^{T} u + q) = ρ b \cdot u + ρ r .$

It splits into the kinetic and internal-energy balances

$\partial_{t} (ρ k) + div (ρ k u - σ^{T} u) = ρ b \cdot u - σ : D,$

$ρ D_{t} e = σ : D - div q + ρ r .$

Thus the boundary power flux is $- σ^{T} u$ and the outward heat flux is $q$ . If additionally $b = - \nabla_{x} Φ$ for a time-independent $C^{1}$ scalar potential $Φ$ defined on an open spatial set containing every $Ω_{t}$ , measured in joules per kilogram, and $r = 0$ , then

$\partial_{t} (ρ (E + Φ)) + div (ρ (E + Φ) u - σ^{T} u + q) = 0.$

Consequently $\int_{C} ρ (E + Φ) d x$ is constant on any bounded fixed $C^{1}$ region $C$ with closure contained in the fluid region throughout $I$ , with no-slip $u = 0$ and insulation $q \cdot n = 0$ , or on a periodic cell $Q = \prod_{j = 1}^{3} [0, L_{j}]$ with side lengths $L_{j} > 0$ metres, $Q \subset Ω_{t}$ for every $t \in I$ , and opposite faces identified by translations when all fields and $Φ$ are periodic. All local equations above are asserted on $O_{B}$ ; the larger $O$ supplies ambient regularity only. These are deductions for any solution with the stated regularity; the theorem does not assert that a solution exists.

Statement references

Tong, Fluid Mechanics, §3.1.2

Powers, Lecture Notes on Intermediate Fluid Mechanics, §§4.4–4.6

Mei, MIT OpenCourseWare, Heat and Energy Conservation, §4.1

Facts & Assumptions

Given: The fields, regularity, signs, SI units, and tensor conventions in the Statement; the smooth continuum balance postulates; the Axiom of Choice; and, for the global conservation conclusion, the selected no-slip/insulating or periodic boundary data.

[A1]

Single-component nonpolar continuum and material balances: material mass, linear-momentum, angular-momentum and total-energy balances hold; the stress traction is $σ n$ , surface power is traction dotted with boundary velocity, heat flux is positive outward, and the specific heat supply contributes $ρ r$ .

[A2]

Specified wall, thermal, periodic and initial data: no-slip and insulating data are jointly selectable for fixed walls, and periodic faces are identified with matching fields and derivative traces; no solvability follows.

[A3]

In the global-conservation case, $b = - \nabla_{x} Φ$ for a time-independent $C^{1}$ scalar potential $Φ$ on an open spatial set containing every $Ω_{t}$ , measured in joules per kilogram, and $r = 0$ ; when using a periodic cell, $Φ$ is periodic.

[F1]

Cauchy stress, local momentum and stress symmetry: under the Axiom of Choice, the stated normalized $C^{2}$ material motion and its induced velocity, positive $C^{1}$ densities, $C^{1}$ stress, continuous body acceleration, and adopted mass, linear- and angular-momentum balances without couples yield local momentum and symmetric stress on the balanced region $O_{B}$ .

[F2]

Material mass balance implies local continuity: the adopted material mass balance supplies the pointwise continuity equation for the smooth positive-density motion.

[F3]

Divergence on a bounded C1 Euclidean domain: under countable choice, the divergence theorem holds on a bounded $C^{1}$ region for a $C^{1}$ vector field on its closure.

[F4]

The Axiom of Choice: every set-indexed family of nonempty sets admits a choice function.

[F5]

Classical frames, primitive quantities and SI units: the SI and classical inertial-frame conventions used by the fields are fixed as in the Statement.

[F6]

Transport across a moving control boundary: under countable choice, a $C^{1}$ scalar field near a bounded $C^{1}$ moving region's swept closure satisfies $\frac{d}{d t} \int_{P_{t}} f = \int_{P_{t}} \partial_{t} f + \int_{\partial P_{t}} f u \cdot n$ when its $C^{2}$ diffeomorphic boundary motion has velocity $u$ .

[F7]

C1 material regions have compatible Jordan volume integrals: diffeomorphisms on neighborhoods of compact closures preserve bounded $C^{1}$ regions and identify open-region integrals with their compact-closure integrals.

Proof

1.1givenF3F4F6algebra

For any countable family of nonempty sets, [F4] provides a choice function on the set of its distinct members; composing with the sequence chooses from each indexed member. Hence the ${A C}_{ω}$ hypothesis used by [F3] and [F6] follows from the stated Axiom of Choice.

2.1givenA1F1F2step 1.1

Apply [F2] to the material mass balance: the Statement supplies the normalized motion, its induced velocity and both positive $C^{1}$ densities. This gives continuity on $O_{B}$ . The same fields and material mass, linear- and angular-momentum balances meet [F1], giving local momentum and symmetric stress there.

2.2givenA1F3F6F7step 1.1algebra

Fix an admissible $V$ and put $P_{t} = χ_{t} (V)$ . By [F7], it is a bounded $C^{1}$ region whose swept closure lies in the ambient field domain. Apply [F6] to $ρ E$ with material boundary velocity $u$ , and [F3] to $ρ E u$ , $σ^{T} u$ and $q$ . Since $(σ n) \cdot u = (σ^{T} u) \cdot n$ , the adopted total-energy balance gives zero integral over every $P_{t}$ of the continuous residual $R : = \partial_{t} (ρ E) + div (ρ E u - σ^{T} u + q) - ρ b \cdot u - ρ r$ . For fixed $t$ , every sufficiently small ball $Q ⋐ Ω_{t}$ is such an image, since $V = χ_{t}^{- 1} (Q)$ is admissible by [F7]. A nonzero value of a continuous $R$ has fixed strict sign on a smaller ball and would have nonzero integral, a contradiction. Thus $R = 0$ throughout $O_{B}$ , proving the stated local total-energy equation and its outward heat-flux sign.

3.1givenF5step 2.1algebra

Put $k = ∣ u ∣^{2} / 2$ . Taking the dot product of $ρ D_{t} u = ρ b + div σ$ from step 2.1 with $u$ and applying the chain rule yields $ρ D_{t} k = ρ b \cdot u + u \cdot div σ$ .

3.2givenA3step 2.1step 2.2algebra

Under [A3], continuity and the chain rule give $\partial_{t} (ρ Φ) + div (ρ Φ u) = ρ D_{t} Φ = ρ u \cdot \nabla Φ$ . Adding this identity to the total-energy equation in step 2.2 cancels the body-power source $ρ b \cdot u$ and, since $r = 0$ , gives the conservative equation for $ρ (E + Φ)$ .

4.1step 2.1step 3.1algebra

By the product rule, $u \cdot div σ = div (σ^{T} u) - σ : D_{x} u$ . Since $σ$ is symmetric and its contraction with the skew part of $D_{x} u$ is zero, $σ : D_{x} u = σ : D$ . The continuity equation from step 2.1 gives $\partial_{t} (ρ k) + div (ρ k u) = ρ D_{t} k$ . Substitution of step 3.1 therefore gives the kinetic-energy balance.

5.1step 2.1step 2.2step 4.1algebra

Subtract the kinetic balance in step 4.1 from the total-energy balance in step 2.2. The identical body-power terms cancel, and the stress-work fluxes cancel, leaving $\partial_{t} (ρ e) + div (ρ e u + q) = σ : D + ρ r$ . Expanding the mass transport term with the continuity equation from step 2.1 gives $ρ D_{t} e = σ : D - div q + ρ r$ .

6.1givenA2A3F3step 1.1step 3.2algebra∎

The $C^{1}$ fields permit differentiation under each fixed-domain integral. For a fixed bounded $C^{1}$ region $C$ with closure inside the fluid region, apply [F3] to the potential-energy equation from step 3.2. The boundary flux $(ρ (E + Φ) u - σ^{T} u + q) \cdot n$ vanishes under no-slip $u = 0$ and insulation $q \cdot n = 0$ . For a periodic cell, the coverage hypothesis $Q \subset Ω_{t}$ ensures that the local equation holds on all of $Q$ at every time. Use the fundamental rectangle $Q = \prod_{j = 1}^{3} [0, L_{j}]$ and put $G = ρ (E + Φ) u - σ^{T} u + q$ . Every component of $G$ is $C^{1}$ and periodic. For each $j$ , the one-dimensional fundamental theorem of calculus gives $\int_{Q} \partial_{x_{j}} G_{j} d x = \int_{\prod_{k \neq j} [0, L_{k}]} (G_{j} (x_{1}, \dots, L_{j}, \dots, x_{3}) - G_{j} (x_{1}, \dots, 0, \dots, x_{3})) d x_{\hat{j}} = 0,$ where $d x_{\hat{j}}$ is the product volume measure in the two coordinates other than $x_{j}$ . Thus $\int_{Q} div G d x = 0$ , and integrating the local equation from step 3.2 gives $\frac{d}{d t} \int_{Q} ρ (E + Φ) d x = 0$ . The periodicity follows from [A2] and [A3].

Kinetic/internal/total energy and boundary work

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources