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Green's theorem is the curl statement for a planar field lifted to $R^{3}$

Statement

Let $D$ be a finite elementary Green region with its supplied decomposition, positively oriented, and let $P, Q$ be $C^{1}$ on an open $U \subseteq R^{2}$ containing $D$ . Define the lift

$\tilde{F} (x, y, z) : = (P (x, y), Q (x, y), 0) ((x, y, z) \in U \times R),$

a field on the open set $U \times R \subseteq R^{3}$ . Then $\tilde{F}$ is $C^{1}$ , its curl has first and second coordinates identically $0$ and third coordinate $\partial_{x} Q - \partial_{y} P$ at every point, independent of $z$ , and the circulation of the planar field around the positive boundary chain equals the integral of the third coordinate of the curl of the lift:

$\int_{\partial D} (P, Q) \cdot d r = \iint_{D} (curl \tilde{F})_{z} (x, y, 0) d A .$

Facts & Assumptions

Given: The finite elementary Green region $D$ with its supplied decomposition and positive orientation, the $C^{1}$ functions $P, Q$ on the open $U \supseteq D$ , and the lift $\tilde{F}$ of the Statement.

[F1]

The curl of a $C^{1}$ field on an open subset of $R^{3}$ is $curl G = (\partial_{y} G_{z} - \partial_{z} G_{y}, \partial_{z} G_{x} - \partial_{x} G_{z}, \partial_{x} G_{y} - \partial_{y} G_{x})$ (Divergence and curl of a $C^{1}$ vector field).

[F2]

A map is of class $C^{k}$ when each component is, a scalar component being $C^{1}$ when its first partial derivatives exist and are continuous ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

For a finite elementary Green region the positive boundary integral is the finite sum over the surviving oriented arcs, and $\int_{\partial D} G \cdot d r$ and $\int_{\partial D} P d x + Q d y$ denote that sum for the field $(P, Q)$ (Positive orientation of elementary-region boundaries, Scalar line integrals with respect to arc length and vector-field line integrals).

[F4]

A finite elementary Green region is a nonempty finite union of elementary Green regions with pairwise disjoint interiors and the stated shared-arc conditions, supplied as data (Type I, Type II, and elementary regions for Green's theorem).

[F5]

For $x, y \in R^{m}$ , $⟨ x, y ⟩ = \sum_{i < m} x_{i} y_{i}$ , and $e_{k}$ has $k$ th coordinate $1$ and the others $0$ (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 $0$ ).

[L1]

Let $D = D_{1} \cup \dots \cup D_{N}$ be a finite elementary Green region with its supplied decomposition, oriented positively, and let $P, Q$ be $C^{1}$ on an open neighbourhood of $D$ . Then $\int_{\partial D} P d x + Q d y = \iint_{D} (\partial_{x} Q - \partial_{y} P) d A$ (Green's theorem for finite unions of elementary regions).

Proof

technique · direct

1.1givenF2F5

The three components of $\tilde{F}$ are $(x, y, z) \mapsto P (x, y)$ , $(x, y, z) \mapsto Q (x, y)$ and the constant $0$ . Their first partial derivatives are $\partial_{x} {\tilde{F}}_{x} = \partial_{x} P$ , $\partial_{y} {\tilde{F}}_{x} = \partial_{y} P$ , $\partial_{z} {\tilde{F}}_{x} = 0$ ; $\partial_{x} {\tilde{F}}_{y} = \partial_{x} Q$ , $\partial_{y} {\tilde{F}}_{y} = \partial_{y} Q$ , $\partial_{z} {\tilde{F}}_{y} = 0$ ; and all three of $\partial_{x} {\tilde{F}}_{z}$ , $\partial_{y} {\tilde{F}}_{z}$ , $\partial_{z} {\tilde{F}}_{z}$ are $0$ . Each of these exists and is continuous on $U \times R$ because $P$ and $Q$ are $C^{1}$ on $U$ , so $\tilde{F}$ is $C^{1}$ there by [F2].

2.1step 1.1F1

By [F1] and step 1.1 the three coordinates of $curl \tilde{F}$ are $\partial_{y} {\tilde{F}}_{z} - \partial_{z} {\tilde{F}}_{y} = 0 - 0 = 0$ , then $\partial_{z} {\tilde{F}}_{x} - \partial_{x} {\tilde{F}}_{z} = 0 - 0 = 0$ , and then $\partial_{x} {\tilde{F}}_{y} - \partial_{y} {\tilde{F}}_{x} = \partial_{x} Q - \partial_{y} P$ . All three are computed, and the third depends only on $(x, y)$ , so its value at $(x, y, z)$ is its value at $(x, y, 0)$ .

3.1step 2.1F3F4L1∎

By [F4] the region $D$ carries its supplied decomposition and $P, Q$ are $C^{1}$ on the open neighbourhood $U$ of $D$ , so [L1] gives $\int_{\partial D} P d x + Q d y = \iint_{D} (\partial_{x} Q - \partial_{y} P) d A$ ; by [F3] the left side is $\int_{\partial D} (P, Q) \cdot d r$ , and by step 2.1 the integrand on the right is $(curl \tilde{F})_{z} (x, y, 0)$ . That is the asserted identity, and step 2.1 is the assertion about the three curl coordinates.

Remarks

This is a dictionary, not a new theorem. Both sides are the two sides of Green's theorem, rewritten. What the corollary records is that the planar integrand $\partial_{x} Q - \partial_{y} P$ is a curl, so that the planar and the spatial developments on this page speak about one operator rather than two unrelated ones.
The route is deliberately one-way. The classical Stokes theorem for a $C^{2}$ patch over a finite elementary Green region is proved from Green's theorem, so re-deriving Green's theorem from it would be circular. Nothing above uses Stokes' theorem.

Depends on

Used by

Dependency tree · two levels

47 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

J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus (University of British Columbia), section 4.3 (standard reference, not scraped)

Green's theorem is the curl statement for a planar field lifted to R3R3