CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Every patch of an elementary solid region's presentation is a graph face in some direction, and at interior base points its normal is outward

Statement

Let $E$ be an elementary solid region with presentation $Σ = ((D_{1}, φ_{1}), \dots, (D_{P}, φ_{P}))$ and sublists $Σ_{k}^{+}, Σ_{k}^{-}, Σ_{k}^{0}$ for $k \in {x, y, z}$ (Elementary solid regions: one boundary presentation adapted in all three coordinate directions). Then every patch of the presentation is an upper or a lower face in at least one coordinate direction: for each $j$ there is $k$ with $j \in Σ_{k}^{+} \cup Σ_{k}^{-}$ .

Moreover, for such a $j$ and $k$ and for every interior parameter point $c \in D_{j}^{\circ}$ whose projection $π_{k} (φ_{j} (c))$ lies in the interior of the base $D_{k}$ of the $k$ th description, the induced unit normal $N_{φ_{j}} (c)$ is the outward unit normal to the tangent plane at $φ_{j} (c)$ .

Facts & Assumptions

Given: The elementary solid region $E$ with its presentation $Σ$ , its three simple descriptions and the three partitions of ${1, \dots, P}$ into sublists.

[F1]

For $j \in Σ_{k}^{0}$ the $k$ th coordinate of $φ_{j, u} \times φ_{j, v}$ vanishes on the interior of $D_{j}$ , and the three sublists $Σ_{k}^{+}, Σ_{k}^{-}, Σ_{k}^{0}$ partition ${1, \dots, P}$ (Boundary presentations adapted to a simple solid region in a coordinate direction, Elementary solid regions: one boundary presentation adapted in all three coordinate directions).

[F2]

A regular patch has $φ_{u} \times φ_{v} \neq 0$ at every point of the interior of its parameter region, and that interior is nonempty (Regular parametrized surface patches on compact Jordan parameter regions).

[F3]

For $v \in R^{3}$ , $⟨ v, e_{k} ⟩ = v_{k}$ , so $v = 0$ if and only if all three of $v_{x}, v_{y}, v_{z}$ are zero (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[F4]

The parametrization induces on the interior the unit normal $N_{φ} = (φ_{u} \times φ_{v}) / ∥ φ_{u} \times φ_{v} ∥_{2}$ (Unit normal fields, orientations, and flux through a regular surface patch).

[F5]

A unit vector $ν$ is outward at $p \in \partial E$ when for some $ε > 0$ one has $p + t ν \notin E$ and $p - t ν \in E$ for every $t$ with $0 < t < ε$ ; with a two-dimensional subspace $T$ supplied, the outward one of its two unit normals is the outward unit normal to $T$ at $p$ (The outward unit normal at a boundary point of a compact solid).

[L1]

Under the hypotheses of an adapted presentation, for $j \in Σ^{+} \cup Σ^{-}$ and an interior parameter point $c$ whose projection lies in the interior of the base, the induced unit normal at $c$ is the outward unit normal to the tangent plane at $φ_{j} (c)$ (At interior base points, the graph faces of an adapted presentation induce the outward unit normal).

Proof

technique · direct

1.1

Fix $j$ and, by [F2], a point $c \in D_{j}^{\circ}$ ; then $φ_{j, u} (c) \times φ_{j, v} (c) \neq 0$ , so by [F3] at least one of its three coordinates is nonzero at $c$ . Fix a direction $k$ for which the $k$ th coordinate is nonzero at $c$ .

givenF2F3

2.1

By [F1], if $j$ belonged to $Σ_{k}^{0}$ then that $k$ th coordinate would vanish at every point of $D_{j}^{\circ}$ , in particular at $c$ , which step 1.1 excludes. The three sublists partition the index set by [F1], so $j \in Σ_{k}^{+} \cup Σ_{k}^{-}$ . This is the first assertion; equivalently, by [F3] and [F4], a patch lateral in all three directions would have an induced unit normal orthogonal to $e_{x}$ , $e_{y}$ and $e_{z}$ and hence equal to $0$ , which no unit vector is.

step 1.1F1F3F4

3.1

Let $j$ and $k$ be as in the second assertion and let $c \in D_{j}^{\circ}$ have $π_{k} (φ_{j} (c))$ in the interior of $D_{k}$ . The presentation is adapted to the $k$ th description by [F1] and $j \in Σ_{k}^{+} \cup Σ_{k}^{-}$ , so [L1] applies and gives that $N_{φ_{j}} (c)$ of [F4] is the outward unit normal to the tangent plane at $φ_{j} (c)$ in the sense of [F5].

step 2.1F1F4F5L1∎

Remarks

The claim is qualified, and the qualification is real. Outwardness is asserted only at interior parameter points whose projection lands in the interior of the relevant base. The excluded points are the parameter-boundary points of a patch and the points sitting over the boundary of the base — the seams and the edges — and at those a normal need not exist or need not be outward. That is not a defect of the presentation: no integral on this page sees a set of content zero in a parameter region.
Why one direction suffices. A patch may be a graph face in one direction and lateral in the other two, as the top face of a box is; the corollary asserts existence of one such direction for each patch, not the same direction for all patches.

Depends on

Used by

The divergence theorem on an elementary solid region Theorem

Dependency tree · two levels

54 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. Corral, Vector Calculus, chapter 4 (LibreTexts) (standard reference, not scraped)