Foundations — Green's theorem — proof sketch, both forms

This page assumes you have seen nothing. We build every symbol the parent note uses — $P$, $Q$, $dx$, $dy$, $\oint$, $\iint$, $\partial$, $\mathbf{n}$, $\mathbf{T}$, $dA$, $ds$, $\cdot$, $\nabla$ — from a blank chalkboard.

0. The playground: points and the plane

Everything happens on a flat sheet, the plane. A location on it is a point, written $(x,y)$.

Why we need it: Green's theorem compares a region (a filled patch of the plane) to its edge. Both are just collections of points $(x,y)$, so this is the alphabet of everything else.

1. A vector field: an arrow at every point

At every point $(x,y)$ we glue a little arrow. This is a vector field.

Picture: think of $\mathbf{F}$ as wind. At each spot the wind blows in some direction with some strength — that's the arrow. $P$ is the eastward wind, $Q$ the northward wind.

Why the topic needs it: Green's theorem is a statement about a vector field. $P$ and $Q$ appear in literally every formula on the parent page. See Line Integrals and Curl and Divergence for what we do with them.

2. The dot product $\mathbf{a}\cdot \mathbf{b}$ — "how much do two arrows agree?"

Before we walk any loops, we need one tiny piece of arrow-arithmetic that the flux form uses.

Why the topic needs it: the flux form writes $\mathbf{F}\cdot \mathbf{n}$ — "how much of the wind $\mathbf{F}$ points along the outward direction $\mathbf{n}$." That number is exactly $F_x{n}_x+F_y{n}_y$, the dot product. Keep this formula handy for §6.

3. Curves, closed loops, and orientation

Why the topic needs it: the parent note's whole left-hand side is "walk around $C$." Get the direction wrong and every equation flips sign — this is one of the listed Common Mistakes. And if $C$ weren't piecewise smooth, the line integral wouldn't even be defined.

4. Region $D$: filled, and hole-free

Picture: $D$ is a floor; $dA$ is one tile in the tiling. Summing over all tiles is the double integral (next section).

5. Integrals: three symbols, one idea (add up tiny bits)

Every integral sign is a stretched "S" for Sum. The differences are what you sum over.

6. Partial derivatives, and the operator $\nabla$

A derivative answers "how fast does this quantity change?" On a 2D function like $P(x,y)$ we must say change in which direction.

Picture: stand on a hilly surface $P$. Walk due East: your uphill steepness is $\partial P/\partial x$. Walk due North: it's $\partial P/\partial y$. Same hill, two different slopes.

The two combinations the parent builds from these partials:

• Curl (scalar): $Q_x-P_y$ — measures local counterclockwise spin.
• Divergence: $P_x+Q_y=\nabla \cdot \mathbf{F}$ — measures local spreading-out.

Both are explained fully in Curl and Divergence; here you only need to know they are combinations of partial derivatives.

7. Tangent $\mathbf{T}$, normal $\mathbf{n}$, arc length $ds$, and "flux"

The flux form needs two special arrows attached to the walk, plus a proper measure of step length.

Why "out"? If the region is on your left (CCW), then turning your forward-arrow to the right (clockwise) makes it point away from the region.

Why the topic needs it: the flux form is literally built from $\mathbf{n}$, $ds$, and the dot product. Everything in it is now defined.

8. The family tree (these all reappear as siblings)

Green's theorem is one floor in a bigger building; Stokes' Theorem and the Divergence Theorem are its 3D relatives, and Conservative Vector Fields are the special case where the loop integral is always $0$. You don't need them yet — just know the vocabulary above unlocks all of them.

How the foundations feed the theorem

Read this bottom-up: each row is built from the ideas above it, and the last row is the theorem itself.

Equipment checklist

Test yourself — if any answer is fuzzy, reread that section.