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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.

Downward flux through the graph $z = x y$ over the unit square

Example

For $F (x, y, z) = (z x, z^{2}, x y)$ , the downward flux through the graph $z = x y$ over $[0, 1]^{2}$ is $- 1 / 18$ . The upward flux is $1 / 18$ .

Facts & Assumptions

Given: The graph function $g (x, y) = x y$ and vector field $F$ .

[L1]

The downward area vector of a $C^{1}$ graph is $(g_{x}, g_{y}, - 1)$ (Surface area, scalar integrals, and flux over a $C^{1}$ graph), and derivative algebra gives $(g_{x}, g_{y}) = (y, x)$ (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 $g (c) \neq 0$ ).

[L2]

Jordan-Fubini and the fundamental theorem evaluate polynomial integrals over the unit square (Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

Verification

technique · direct

1.1

On the graph, $F (x, y, x y) = (x^{2} y, x^{2} y^{2}, x y)$ , and its dot product with the downward area vector $(y, x, - 1)$ is $x^{2} y^{2} + x^{3} y^{2} - x y$ .

givenL1algebra

2.1

By [L2], the integral is $1 / 9 + 1 / 12 - 1 / 4 = - 1 / 18$ .

step 1.1L2algebra

3.1

Reversing the orientation negates flux, so the upward value is $1 / 18$ .

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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

University of Toronto MAT237 notes, Section 5.3, Basic Problem 3 (standard reference, not scraped)

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