Objectives
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Apply Green’s theorems to compute line integrals and planar areas.
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Relate circulation and flux to curl and divergence in the plane.
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Interpret multiply connected regions and orientation when applying the theorems.
Section 20.2 Green’s theorems
Introduction goes here.
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Motivation: What’s the relationship between circulation, flux, curl, and divergence?
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Theorem 12. Green’s theorems.
Let \(C\) be a positively-oriented, piecewise-smooth, simple closed curve in the plane, and let \(D\) be the region bounded by \(C\text{.}\) Furthermore, let \(\bv F=(P,Q)\) be a vector field such that \(P\) and \(Q\) have continuous partial derivatives on an open region that contains \(D\text{.}\) Then:-
Green’s curl theorem:\begin{align*} \oint_C \bv{F}\cdot\dd{\bv{r}} = \int_C \bv{F}\cdot\uv{t}\dd{s} \amp = \iint_D \Curl\bv{F}\dd{A}\\ \oint_C P\dd{x} + Q\dd{y} \amp = \iint_D \qty(\pdv{Q}{x} - \pdv{P}{y})\dd{A} \end{align*}
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Green’s divergence theorem:\begin{align*} \int_C \bv{F}\cdot\uv{n}\dd{s} = \iint_D \Div\bv{F}\dd{A}\\ \oint_C P\dd{y} - Q\dd{x} \amp = \iint_D \qty(\pdv{P}{x} + \pdv{Q}{y})\dd{A} \end{align*}
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Area of \(D\text{:}\)\begin{equation*} A=\oint_{C} x\dd{y}=-\oint_{C} y\dd{x}=\frac{1}{2}\oint_{C} x\dd{y}-y\dd{x} \end{equation*}
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Multiply connected domain:\begin{equation*} \iint_D \Curl\bv{F}\dd{A} = \oint_{C\tsub{outside}}\bv{F}\cdot\dd{\bv{r}} - \oint_{C\tsub{inside}}\bv{F}\cdot\dd{\bv{r}} \end{equation*}
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Can think of divergence as flux per unit infinitesimal area and curl as circulation per unit infinitesimal area.
