MATH 251 Lecture 32

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Green's Theorem

(See Vector Analysis Theorems#Green's Theorem→)

When you add up all these little swirly thingies, you get the big swirly thingy.

For ${\vec {F}}(x,y)=P(x,y){\hat {\imath }}+Q(x,y){\hat {\jmath }}$ in a region $\Omega$ bounded by the curve $C$,

$\oint _{C}{\vec {F}}\cdot \mathrm {d} {\vec {r}}=\iint _{\Omega }{\vec {\nabla }}\times {\vec {F}}\cdot {\hat {k}}=\iint _{\Omega }{\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\,\mathrm {d} A$

This could be an elaborate way to calculate area when the curl is equal to 1.

Example

Find $\int _{C}x\sin {y}\,\mathrm {d} x-y\,\mathrm {d} y$, where $C$ is the upper half of the unit circle.

$P(x,y)=x\sin {y}$ and $Q(x,y)=-y$.

${\begin{aligned}Q_{x}-P_{y}&=-x\cos {y}\\\int _{-1}^{1}\int _{0}^{\sqrt {1-x^{2}}}-x\cos {y}\,\mathrm {d} y\,\mathrm {d} x&=\int _{-1}^{1}x\sin {\left({\sqrt {1-x^{2}}}\right)}\,\mathrm {d} x\\&=0\end{aligned}}$

Because $x\sin {\left({\sqrt {1-x^{2}}}\right)}$ is odd, and we integrate it from -1 to 1.

Conservative Vector Fields

We have ${\vec {F}}$ that is known to be conservative. We know that $\oint _{C}{\vec {F}}\cdot \mathrm {d} {\vec {r}}=0$ over every loop.

By Green's Theorem, this also means that $\oint _{C}{\vec {F}}\cdot \mathrm {d} {\vec {r}}=\iint _{R}{\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\,\mathrm {d} A=0$

In order for $\iint _{R}{\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\,\mathrm {d} A=0$ to be true, the integrand $Q_{x}-P_{y}=0$, which also means that $Q_{x}=P_{y}$

Let's find a potential function $f(P)$ that starts at a base point (0,0) and connects to $P$ such that $f(P)=\oint _{C}{\frac {F}{\cdot }}\mathrm {d} {\vec {r}}$ where $C$ connects the two points and could be any path.

Fact: ${\frac {\partial f}{\partial x}}=P$ and ${\frac {\partial f}{\partial y}}=Q$.

Stoke's Theorem

(See Vector Analysis Theorems#Stokes' (Curl) Theorem→)

On a 3D surface, there are many swirlies, but the swirlies may not lie in the $xy$ plane. Let ${\hat {n}}$ represent the unit vector normal to the surface at a point on the surface.

$\iint {\vec {\nabla }}\times {\vec {F}}\cdot {\hat {n}}\,\mathrm {d} S=\int _{C}{\vec {F}}\cdot \mathrm {d} {\vec {r}}$