Green, Gauss, and Stokes

Three fundamental theorems relate vector differential calculus and vector integral calculus

Green’s Theorem

Suppose $D$ is a $y$-simple region and let $C$ be its boundary. Suppose $P:D\to \mathbb{R}$ is of class $C^1$.
$D$ can be described by $a\le x\le b$, ${\varphi }_1(x)\le y\le {\varphi }_2(x)$
In this notation, $C^{+}$ refers to the counterclockwise path of $C$, while $C^{-}$ refers to the clockwise path
$C^{+}=C_1^{+}+B_2^{+}+C_2^{-}+B_1^{-}$ (where $B$ are vertical lines)

${\iint }_D\frac{\partial P}{\partial y}(x,y)dxdy={\int }_a^b{\int }_{{\varphi }_1(x)}^{{\varphi }_2(x)}\frac{\partial P}{\partial y}(x,y)dydx={\int }_a^b[P(x,{\varphi }_2(x))-P(x,{\varphi }_1(x))]dx$
Since $C_1^{+}$ can be parameterized as $x\mapsto (x,{\varphi }_1(x))$, $a\le x\le b$, the integral simplifies to a line integral
$=-{\int }_{C_1^{+}}Pdx-{\int }_{C_2^{-}}Pdx$
$=-{\int }_{C_1^{+}}Pdx-{\int }_{C_2^{-}}Pdx+{\int }_{B_1^{-}}Pdx+{\int }_{B_2^{+}}Pdx$ (since $B$ does not change across $x$)
$=-{\int }_{C^{+}}Pdx$

Lemma 1: ${\int }_{C^{+}}Pdx=-{\iint }_D\frac{\partial P}{\partial y}(x,y)dxdy$

Equivalent logic with the roles of $x$ and $y$ swapped yields,
Lemma 2: ${\int }_{C^{+}}Qdy={\iint }_D\frac{\partial Q}{\partial x}(x,y)dxdy$

Green’s Theorem: ${\int }_{C^{+}}Pdx+Qdy={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy$

Green’s theorem relates a line integral around a region to the area within the region. Note that more complex regions can be split into simple regions on which Green’s theorem applies.

This can be used to find the area surrounded by a curve,
Area of a Curve: $A=\frac{1}{2}{\int }_{\partial D}xdy-ydx$

Vector Form of Green’s Theorem: ${\int }_{\partial D}\mathbf{F}\cdot d\mathbf{s}={\iint }_D(\text{curl}\mathbf{F})\cdot \mathbf{k}dA={\iint }_D(\nabla \times \mathbf{F})\cdot \mathbf{k}dA$
This vector form uses the concept of curl, $(\nabla \times \mathbf{F})\cdot \mathbf{k}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$
Note that $\mathbf{k}$ is a unit vector

Divergence Form: ${\int }_{\partial D}\mathbf{F}\cdot \mathbf{n}ds={\iint }_D\text{div }\mathbf{F}dA$, where $\mathbf{n}=\frac{y^{\prime }(t)-x^{\prime }(t)}{\sqrt{(x^{\prime }(t){)}^2+(y^{\prime }(t){)}^2}}$ is the outward unit normal to $\partial D$
This uses the concept of divergence, $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}$ and $\text{div }\mathbf{F}=\frac{\partial P}{dx}+\frac{\partial Q}{dy}$

Stokes’ Theorem

Consider a surface $S$ parameterized by $(x,y,f(x,y))$. Suppose $c:[a,b]\to {\mathbb{R}}^2=(x(t),y(t))$ is a parameterization of $\partial D$ in the positive direction (the direction that validates Green’s theorem, counterclockwise in simple curves).

The boundary curve $\partial S$ is the image of the mapping $p:t\to (x(t),y(t),f(x(t),y(t)))$

Stokes’ Theorem: ${\iint }_S\text{curl }\mathbf{F}\cdot d\mathbf{S}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}={\int }_{\partial S}\mathbf{F}\cdot d\mathbf{s}$
This says essentially that the integral of the normal component of the curl of $F$ over $S$ is equal to the integral of the tangential component of $F$ around the boundary of $S$

Note the useful formula ${\iint }_{S}F\cdot dS={\iint }_D\left[F_1\left(-\frac{\partial z}{\partial x}\right)+F_2\left(-\frac{\partial z}{\partial y}\right)+F_3\right]dxdy$ which helps transform an expression like the lefthand side of Stokes’ theorem

Suppose $\Phi :D\to {\mathbb{R}}^3$ is a parameterization of $S$
In the general case, it would be mistaken to define $\partial S$ as $\Phi (\partial D)$; For one $S$ need not even have an edge, it could be a closed surface! $\Phi$ must be restricted to be one-to-one

With this definition in mind, Stokes’ theorem also works on parameterized surfaces. If the surface has no geometric boundary, then ${\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}=0$

Stokes’ theorem can be used to justify interpreting curl as circulation per unit area
Some manipulation derives the formula $\text{curl }\mathbf{V}(P)\cdot \mathbf{n}={\lim }_{\rho \to 0}\frac{1}{A(S_{\rho })}{\int }_{\partial S_{\rho }}\mathbf{V}\cdot d\mathbf{s}$, where $S_{\rho }$ is a disc of radius $\rho$ centered at $P$

Now when $V$ is the velocity field of a fluid, ${\int }_C\mathbf{V}\cdot d\mathbf{s}$ clearly represents the circulation around $C$ (if $V$ is tangent to the curve then there is no circulation). This means that $\text{curl }\mathbf{V}(P)\cdot \mathbf{n}$ represents the circulation of $V$ per unit area at $P$ on a surface (perpendicular to $\mathbf{n}$)

In this context, $\text{curl }\mathbf{V}$ is often called the vorticity vector

Conservative Fields

Some fields have a useful property that they can be expressed gradients, such that ${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}=f(\mathbf{c}(b))-f(\mathbf{c}(a))$, since $\mathbf{F}=\nabla f$

Definition: A conservative vector field $\mathbf{F}$ defined on ${\mathbb{R}}^3$ except for a finite amount of points satisfies the following equivalent properties:

1. On any simple closed curve ${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}=0$
2. ${\int }_{{\mathbf{C}}_1}\mathbf{F}\cdot d\mathbf{s}={\int }_{{\mathbf{C}}_2}\mathbf{F}\cdot d\mathbf{s}$ if $C_1$ and $C_2$ have the same endpoints
3. $\mathbf{F}$ is the gradient of some function $f$ (and $f$ is undefined where $\mathbf{F}$ is undefined)
4. $\nabla \times \mathbf{F}=0$

Physically, ${\int }_{\mathbf{c}}\mathbf{F}\cdot d\mathbf{s}$ can be thought of as representing the work done by $F$ in moving along a path, or it can be thought of as the amount of circulation across the path. If $\text{curl }\mathbf{F}=\nabla \times \mathbf{F}=0$ then the field is irrotational, meaning $\mathbf{F}$ is a gradient field for some function $f$ (which is then called a potential for $\mathbf{F}$)

In ${\mathbb{R}}^2$, $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}$ means $\nabla \times \mathbf{F}=0$ reduces to $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ as long as $\mathbf{F}$ is smooth everywhere

Theorem: If $\mathbf{F}$ is a $C^1$ vector field on ${\mathbb{R}}^3$ with $\text{div }\mathbf{F}=0$ then there exists a $C^1$ field $\mathbf{G}$ with $\mathbf{F}=\text{curl }\mathbf{G}$

Gauss’ Theorem

Closed surfaces can generally be oriented inwards or outwards, and if they are oriented outwards ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$ measures the total outwards flux

Gauss’ Divergence Theorem: Let $W$ be a symmetric elementary region in space (can be written as an elementary region on any axis) and $\partial W$ be the oriented closed surface bounding the volume, then ${\iiint }_W(\nabla \cdot \mathbf{F})dV={\iint }_{\partial W}\mathbf{F}\cdot d\mathbf{S}$

Alternatively, ${\iiint }_W(\text{div }\mathbf{F})dV={\iint }_{\partial W}(\mathbf{F}\cdot \mathbf{n})dS$

The proof in the book is nice, and pretty similar to that of Green’s theorem
Gauss’ theorem can be applied to more complex regions that can be broken up into symmetric elementary regions

In a process similar to what was done with analyzing curl, divergence can be shown to represent the rate of net outward flux at $P$ per unit volume; If $\text{div }\mathbf{F}(P)>0$ then $P$ is called a source, otherwise if $\text{div }\mathbf{F}(P)<0$ then $P$ is called a sink

If $\text{div }\mathbf{F}(P)=0$ then $\mathbf{F}$ is divergence-free which also implies ${\iint }_S\mathbf{F}\cdot d\mathbf{S}=0$
A fluid with this property is incompressible

Gauss’ Law: Let $M$ be a symmetric elementary region in ${\mathbb{R}}^3$, if $(0,0,0)\notin \partial M$ then ${\iint }_{\partial M}\frac{\mathbf{r}\cdot \mathbf{n}}{r^3}dS=\{\begin{array}{l}4\pi \text{if }(0,0,0)\in M\\ 0\text{ if }(0,0,0)\notin M\end{array}$
where $\mathbf{r}(x,y,z)=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ and $r(x,y,z)=\Vert \mathbf{r}(x,y,z)\Vert$

The proof for this involves looking at the region between $M$ and a ball

Gauss’ theorem can also be used to derive $\text{div }\mathbf{F}=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }({\rho }^2{F}_{\rho })+\frac{1}{\rho \sin \varphi }\frac{\partial }{\partial \varphi }(\sin \varphi F_{\varphi })+\frac{1}{\rho \sin \varphi }\frac{\partial F_{\theta }}{\partial \theta }$

It is also very much related to Maxwell’s equations

Differential Forms

The theory of differential forms formulates Green’s, Stokes’, and Gauss’ theorems as a single statement. The question is how to generalize ${\int }_a^b{f}^{\prime }(x)dx=f(b)-f(a)$ to higher dimensions. The cross product does not generalize to higher dimensions either.

Definition: A differential form is a new mathematical object. $Pdx+Qdy$ is an example of a 1-form, while $Fdxdy$ is a 2-form. The operation $d$ takes $n$-forms to $n+1$-forms.
This is like a generalized curl where if $\omega =Pdx+Qdy$ then $d\omega =\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy$

The Fundamental Theorem of Calculus (generalized Stokes’ theorem): ${\int }_{\partial M}\omega ={\int }_{M}d\omega$

Note: Researching this topic is difficult, but I’ve gathered that the definitions provided here are unnecessarily restricted to 3 dimensions and below.

Now, to back up a bit…
Definition: Let $K$ be an open set in ${\mathbb{R}}^3$, a 0-form on $K$ is a real-valued function $f:K\to \mathbb{R}$. When $f$ is differentiated once, it is assumed to be in class $C^1$, and $C^2$ if differentiated twice. 0-forms can be added or multiplied together.

Definition: The basic 1-forms are $dx$, $dy$, and $dz$. A 1-form $\omega$ on $K$ is a linear combination $\omega =Pdx+Qdy+Rdz$. These can be added together, or multiplied by 0-forms.

Definition: The basic 2-forms are $dxdy$, $dydz$, and $dzdx$. A 2-form $\eta$ on $K$ is a linear combination $\eta =Fdxdy+Gdydz+Hdzdx$. These can be added together, or multiplied by 0-forms.

It’s not immediately clear what these forms are good for.

Let $\omega =Pdx+Qdy+Rdz$, notice how ${\int }_C\omega ={\int }_a^b\dots$ effectively associates a real number to each curve $C\subset K$

Likewise in higher dimensions, a 2-form assigns each real number to a surface. Let $\eta =Fdxdy+Gdydz+Hdzdx$, then ${\iint }_S\eta$ is the same regardless of specific parameterization $\Phi (u,v)$

Finally, for the purposes of these notes, it’s only necessary to note that for 3-forms, $v=fdxdydz$ means ${\iiint }_{R}v={\iiint }_{R}fdxdydz$ which is just a regular triple integral

Definition: The wedge product $\omega \wedge \eta$ operates on a $k$-form and an $l$-form $0\le k+l\le 3$ and creates a $k+l$ form, satisfying the following laws ($f$ is a 0-form),

1. For each $k$ there is a zero $k$-form 0 where $0+\omega =\omega$ and $0\wedge \eta =0$ if $0\le l\le 3$ (that last clause is phrased weirdly in the textbook)
2. $(f{\omega }_1+{\omega }_2)\wedge \eta =f({\omega }_1\wedge \eta )+({\omega }_2\wedge \eta )$
3. $\omega \wedge \eta =(-1{)}^{kl}(\eta \wedge \omega )$
4. If ${\omega }_1,{\omega }_2,{\omega }_3$ are $k_1,k_2,k_3$ forms respectively and $k_1+k_2+k_3\le 3$ then ${\omega }_1\wedge ({\omega }_2\wedge {\omega }_3)=({\omega }_1\wedge {\omega }_2)\wedge {\omega }_3$
5. $\omega \wedge (f\eta )=(f\omega )\wedge \eta =f(\omega \wedge \eta )$
6. The following rules: $dx\wedge dy=dxdy$, $dx\wedge dx=0$, $dx\wedge (dy\wedge dz)=dxdydz$
7. $f\wedge \omega =f\omega$

Definition: The derivative of a 3-form is always 0. Otherwise, the derivative of a $k$-form is a $k+1$-form, following the following rules ($f$ is a 0-form),

1. $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$
2. $d({\omega }_1+{\omega }_2)=d{\omega }_1+d{\omega }_2$
3. If $\omega$ is a $k$-form, $d(\omega \wedge \eta )=(d\omega \wedge \eta )+(-1{)}^k(\omega \wedge d\eta )$
4. $d(d\omega )=0$ or put in other words $d^2=0$

Now, Green’s theorem would say, let $D$ be an elementary region in the $xy$ plane, with $\partial D$ given counterclockwise orientation. Suppose $\omega =P(x,y)dx+Q(x,y)dy$ is a 1-form on some open set $K$ in ${\mathbb{R}}^3$ that contains $D$, then ${\int }_{\partial D}\omega ={\iint }_{D}d\omega$

Stokes’ theorem looks quite similar: Let $S$ be an oriented surface in ${\mathbb{R}}^3$ with boundary consisting of a simple closed curve $\partial D$, suppose $\omega$ is a 1-form then ${\int }_{\partial S}\omega ={\iint }_{S}d\omega$

And Gauss’ theorem says ${\iint }_{\partial W}\eta ={\iiint }_{W}d\eta$

Don’t these look similar! In the vector-field formulations, we used divergence for regions in ${\mathbb{R}}^3$ and curl for curves in ${\mathbb{R}}^3$ and regions in ${\mathbb{R}}^2$.

Definition: An oriented 2-manifold with boundary is a surface in ${\mathbb{R}}^3$ whose boundary is a simple closed curve and an oriented 3-manifold with boundary is an elementary region in ${\mathbb{R}}^3$ whose boundary is a surface

Generalized Stokes’ Theorem: Let $M$ be an oriented $k$-manifold in ${\mathbb{R}}^3$ contained in some open set $K$. Suppose $\omega$ is a $(k-1)$-form on $K$. Then ${\int }_{\partial M}\omega ={\iint }_{M}d\omega$