TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Parallel-axis theorem: the inertia tensor about a displaced point of a centred mass distribution

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), used only to make the displayed integrals finite. Let $ν$ be a mass distribution on $R^{3}$ with total mass $m = ν (R^{3}) \in (0, + \infty)$ that is centred, that is, $μ (ν) = \int ρ d ν (ρ) = 0 \in R^{3}$ (Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix), and let $I (ν)$ be its inertia matrix. Then for every $a \in R^{3}$ the inertia tensor of $ν$ about $a$ (The inertia tensor of a finite bounded Borel measure about any point of $R^{3}$ ) satisfies

$I_{ν} (a) = I (ν) + m (∥ a ∥_{2}^{2} I_{3} - a a^{T}),$

where $∥ \cdot ∥_{2}$ is the Euclidean norm, $I_{3}$ the identity matrix and $a a^{T}$ the matrix product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations and Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes. Entrywise,

$I_{ν} (a)_{j k} = I (ν)_{j k} + m (∥ a ∥_{2}^{2} δ_{j k} - a_{j} a_{k}) (j, k \in {1, 2, 3}) .$

Facts & Assumptions

Given: A centred mass distribution $ν$ with total mass $m$ , a point $a \in R^{3}$ , and the notation of Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix and The inertia tensor of a finite bounded Borel measure about any point of $R^{3}$ : the first moment $μ (ν)$ , the inertia matrix $I (ν)$ with $I (ν)_{j k} = \int (∥ ρ ∥_{2}^{2} δ_{j k} - ρ_{j} ρ_{k}) d ν$ , and the inertia tensor $I_{ν} (a)$ with $I_{ν} (a)_{j k} = \int (∥ ρ - a ∥_{2}^{2} δ_{j k} - (ρ_{j} - a_{j}) (ρ_{k} - a_{k})) d ν$ .

[L1]

Finiteness and integrability (Finiteness and a uniform bound for the entries of the inertia tensor about an arbitrary point, Finiteness of the moments of a bounded finite Borel measure on $R^{3}$ , Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix): the functions $ρ \mapsto ρ_{j}$ , $ρ \mapsto ρ_{j} ρ_{k}$ , and the shifted polynomials $∥ ρ - a ∥_{2}^{2} δ_{j k} - (ρ_{j} - a_{j}) (ρ_{k} - a_{k})$ are $ν$ -integrable; the entries $I_{ν} (a)_{j k}$ , $I (ν)_{j k}$ and $μ (ν)_{j}$ are finite real numbers.

[L2]

Linearity of the Lebesgue integral on $L^{1}$ (The Lebesgue integral is linear on $L^{1} (μ)$ , Integrable real and complex functions, and their integrals, Integral over a measurable subset): for real $α, β$ and integrable real functions $f, g$ , the combination $α f + β g$ is integrable and $\int (α f + β g) d ν = α \int f d ν + β \int g d ν$ .

[L3]

Centredness and total mass (Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix, The nonnegative Lebesgue integral, The integral of a nonnegative simple function): $\int ρ_{j} d ν = μ (ν)_{j} = 0$ for each $j$ , and the constant function $1$ has $\int 1 d ν = ν (R^{3}) = m$ .

[L4]

Component algebra (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes): for all $ρ, a \in R^{3}$ and all $j, k$ , $∥ ρ - a ∥_{2}^{2} = ∥ ρ ∥_{2}^{2} - 2 ⟨ ρ, a ⟩ + ∥ a ∥_{2}^{2}, (ρ_{j} - a_{j}) (ρ_{k} - a_{k}) = ρ_{j} ρ_{k} - ρ_{j} a_{k} - a_{j} ρ_{k} + a_{j} a_{k},$ and $⟨ ρ, a ⟩ = \sum_{l} a_{l} ρ_{l}$ .

Proof

Proof technique: direct; expand the shifted integrand into the parent integrand plus constants and linear terms, and integrate termwise, the linear terms vanishing by centredness.

1.1givenL4algebra

Expansion of the integrand: for every $ρ \in R^{3}$ and all $j, k$ , [L4] gives $∥ ρ - a ∥_{2}^{2} δ_{j k} - (ρ_{j} - a_{j}) (ρ_{k} - a_{k}) = [∥ ρ ∥_{2}^{2} δ_{j k} - ρ_{j} ρ_{k}] + [∥ a ∥_{2}^{2} δ_{j k} - a_{j} a_{k}] + [- 2 ⟨ ρ, a ⟩ δ_{j k} + ρ_{j} a_{k} + a_{j} ρ_{k}] .$

2.1step 1.1givenL1L2

Integrability of the pieces: the first bracket is integrable by [L1]; the second bracket is a constant function and hence integrable because $ν (R^{3}) < + \infty$ ; the functions $ρ \mapsto ρ_{j}$ are integrable by [L1], so the third bracket is a finite real linear combination of integrable functions and is integrable by [L2].

3.1step 1.1step 2.1L1L2L3algebra

Termwise integration: applying [L2] to the decomposition of step 1.1 and using the defining integrals of [L1] and the values of [L3], $I_{ν} (a)_{j k} = I (ν)_{j k} + m (∥ a ∥_{2}^{2} δ_{j k} - a_{j} a_{k}) - 2 δ_{j k} \sum_{l} a_{l} μ (ν)_{l} + a_{k} μ (ν)_{j} + a_{j} μ (ν)_{k},$ and every term of the last group vanishes because $μ (ν) = 0$ .

4.1step 3.1L4algebra∎

Conclusion in matrix form: reading the entrywise identity of step 3.1 as a matrix identity gives $I_{ν} (a) = I (ν) + m (∥ a ∥_{2}^{2} I_{3} - a a^{T})$ , which is the statement.

Remarks

Centredness is exactly what is used. The proof shows that for a general mass distribution the same expansion leaves the additional terms $- 2 (⟨ a, μ (ν) ⟩) δ_{j k} + a_{k} μ (ν)_{j} + a_{j} μ (ν)_{k}$ ; the centred case is the classical parallel-axis form, and the theorem asserts nothing about the non-centred case beyond this displayed expansion.
Shape independence. The derivation uses only the finiteness of the first and second moments and the vanishing of the first moment, so the conclusion covers positive-total-mass point, line, surface and volume distributions alike: $ν$ is an arbitrary bounded finite Borel measure of positive total mass, as in Mass distributions on $R^{3}$ : total mass, first moment and inertia matrix.

Parallel-axis theorem: the inertia tensor about a displaced point of a centred mass distribution

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources