TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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Surface area, scalar integrals, and flux over a $C^{1}$ graph

Statement

Let $D \subseteq R^{2}$ be a compact Jordan parameter region and let $g$ be $C^{1}$ on a neighbourhood of $D$ . For the graph $S = {(x, y, g (x, y)) : (x, y) \in D}$ and every continuous real-valued function $q$ on $S$ , $Area (S) = \int_{D} \sqrt{1 + ∥ \nabla g ∥_{2}^{2}}, \int_{S} q d S = \int_{D} q (x, y, g (x, y)) \sqrt{1 + ∥ \nabla g ∥_{2}^{2}} .$ For a continuous vector field $F$ on $S$ , upward flux is $\int_{D} F (x, y, g (x, y)) \cdot (- g_{x}, - g_{y}, 1) .$ For the graph of $g$ over $D$ , the downward flux is the negative of $\int_{D} F (x, y, g (x, y)) \cdot (- g_{x}, - g_{y}, 1)$ .

Facts & Assumptions

Given: The region $D$ , function $g$ , graph parametrization $φ (x, y) = (x, y, g (x, y))$ , continuous scalar field $q$ , and continuous vector field $F$ .

[L1]

Derivative algebra and the gradient definition give $φ_{x} = (1, 0, g_{x})$ and $φ_{y} = (0, 1, g_{y})$ (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[L2]

A nonzero parameter cross product gives a regular patch, area density is its norm, and scalar integrals and flux use the corresponding parameter integrands (Regular parametrized surface patches on compact Jordan parameter regions, The surface area density is the norm of the cross product of the parameter tangents, Surface area and scalar surface integrals on a regular patch, Unit normal fields, orientations, and flux through a regular surface patch).

Proof

technique · direct

1.1

By [L1], $φ_{x} \times φ_{y} = (- g_{x}, - g_{y}, 1)$ , whose norm is $\sqrt{1 + g_{x}^{2} + g_{y}^{2}} = \sqrt{1 + ∥ \nabla g ∥_{2}^{2}}$ and which never vanishes. The first two coordinates make $φ$ injective, so it is a regular patch by [L2].

givenL1L2algebra

2.1

Substituting the norm from step 1.1 into the area and scalar-integral definitions in [L2] gives the first two formulas.

step 1.1L2

2.2

Retaining the signed vector from step 1.1 in the flux definition gives the upward formula; the downward orientation uses its negative and therefore negates the integral.

step 1.1L2

3.1

These substitutions establish all four displayed formulas, including the orientation distinction.

step 2.1step 2.2∎

Depends on

Used by

Downward flux through the graph z=xy over the unit square Example

Dependency tree · two levels

28 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 3.2.21-3.2.23 (standard reference, not scraped)
University of Toronto MAT237 notes, Section 5.3, Special Cases (standard reference, not scraped)

Surface area, scalar integrals, and flux over a C1C1 graph