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The surface area density is the norm of the cross product of the parameter tangents
Statement
For every parameter point, .
The common value is positive in the interior of a regular patch; it may vanish on the parameter boundary under the admitted seam and endpoint convention.
Facts & Assumptions
Given: A regular parametrized surface patch .
The density is the nonnegative square root of the determinant of the Gram matrix of (The first fundamental form, Gram matrix, and area density of a surface patch).
The determinant of a two-vector Gram matrix equals the squared cross-product norm (The squared cross-product norm is the Gram determinant of two vectors).
Proof
By [L1] and [L2], .
Both sides of the claimed equality are nonnegative, so uniqueness of the nonnegative square root gives .
Regularity makes the cross product nonzero in the interior, while the patch definition permits boundary zeros; this proves the qualification.
Depends on
Used by
- Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
- Unit normal fields, orientations, and flux through a regular surface patch Definition
- A closed cylinder as a finitely patched oriented surface Example
- Opposite parametrizations preserve area and negate flux Example
- Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
- The surface area of a torus is 4π²ab Example
- The oriented area vector transforms by the parameter Jacobian determinant Lemma
- Scalar surface integrals on a surface of revolution Theorem
- Surface area, scalar integrals, and flux over a C¹ graph Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- M. E. Taylor, Introduction to Analysis in Several Variables, formulas 3.2.18-3.2.20 (standard reference, not scraped)
- R. Sjamaar, Manifolds and Differential Forms, Theorem 8.4 (standard reference, not scraped)