ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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Oriented area and volume are recovered from wedges and Gram determinants

Example

In $R^{2}$ with the standard inner product, the parallelogram spanned by $u = (1, 2)$ and $v = (3, 1)$ has Gram matrix $(\begin{array}{cc} 5 & 5 \\ 5 & 10 \end{array})$ , whose determinant is $25$ , so its area is $5$ . In $R^{3}$ , the parallelotope spanned by $u = (1, 0, 1)$ , $v = (0, 2, 0)$ , $w = (1, 1, 0)$ has Gram matrix $(\begin{array}{ccc} 2 & 0 & 1 \\ 0 & 4 & 2 \\ 1 & 2 & 2 \end{array})$ , whose determinant is $4$ , so its volume is $2$ .

Facts & Assumptions

Given: The standard inner products on $R^{2}$ and $R^{3}$ and the displayed vectors.

[L1]

The Gram formula gives $∥ v_{1} \land \dots \land v_{k} ∥^{2} = \det G (v_{1}, \dots, v_{k})$ on the exterior power (The Gram formula gives a well-defined positive-definite inner product on exterior powers, and $∥ v_{1} \land \dots \land v_{k} ∥^{2}$ is the Gram determinant).

[L2]

The oriented unit area/volume form has norm $1$ in the Gram pairing, so the norm of a pure wedge is the unoriented area or volume of the spanned parallelepiped (The oriented unit volume form).

Verification

technique · direct

1.1

By [L1], $∥ u \land v ∥^{2} = \det (\begin{array}{cc} ⟨ u, u ⟩ & ⟨ u, v ⟩ \\ ⟨ v, u ⟩ & ⟨ v, v ⟩ \end{array}) = \det (\begin{array}{cc} 5 & 5 \\ 5 & 10 \end{array}) = 25$ , so the parallelogram area is $\sqrt{25} = 5$ .

L1algebra

1.2

By [L1], $∥ u \land v \land w ∥^{2} = \det (\begin{array}{ccc} 2 & 0 & 1 \\ 0 & 4 & 2 \\ 1 & 2 & 2 \end{array}) = 2 (8 - 4) + (0 - 4) = 4$ , so the parallelotope volume is $\sqrt{4} = 2$ .

L1algebra

2.1

By [L2], the unit volume forms have norm $1$ , so the norms computed in steps 1.1 and 1.2 are precisely the unoriented area and volume; the orientation data would only attach a sign, not a size.

L2step 1.1step 1.2

3.1

Steps 1.1, 1.2 and 2.1 recover area and volume from wedges and Gram determinants.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Reyer Sjamaar, Manifolds and Differential Forms, §8.1 (standard reference, not scraped)