The Fundamental Theorem of Calculus said that an integral over an interval is decided by its two endpoints. The boundary controlled the inside.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Green's theorem
The circulation of a field around a closed curve equals a double integral of its curl over the region inside.
Curl
Curl measures the local spin of a field: how fast a tiny paddle wheel would turn if dropped at that point.
Why it helps
It swaps a hard boundary integral for an easier area one, or the other way around. Choose whichever side is simpler.
Area from the boundary
Chosen well, the theorem computes area by walking only the boundary. That is exactly how a planimeter works.
What comes after
Stokes' theorem lifts Green's into three dimensions, and the Divergence theorem does the same for flux. All three say the boundary knows the interior.
The boundary knows the inside
Green's theorem equates a line integral around a closed curve with a double integral over the region it encloses. Information about the interior is entirely encoded in the boundary.
It converts hard into easy
A difficult line integral can become a straightforward double integral, or the reverse. Choosing which side to compute is the practical use of the theorem.
Orientation is part of the hypothesis
The curve must be traversed anticlockwise for the theorem as usually stated. Reversing the orientation flips the sign, so the direction is not a detail.
One of a family
Green's, Stokes' and the divergence theorem all say that an integral over a region equals one over its boundary. They generalise the fundamental theorem of calculus, which said the same for an interval.
Step 2: Try It Yourself
Tap and try it out.
- Released from(2, 0)
Every arrow is perpendicular to the line from the origin, so trajectories circle rather than approach.
Arrows are horizontal, with a size set by height. Nothing moves vertically at all.
Step 3: Watch an Example
One step at a time.
Watch Priya Choose the Easier Side
Priya must find the circulation of a field with constant curl 2 around a circle of radius 3.
- Step 1
She sees that walking the boundary directly would need a parametrised line integral.
Step 4: Your Turn
Practice makes it stick.
The Circulation
Problem 1 of 2
A constant curl of 3 over a region of area 10. What is the circulation around the boundary?
The Still Field
Problem 2 of 2
A field with curl zero everywhere. What is the circulation around any closed loop?
Boundary and Interior
1 of 8
A constant curl of 5 over an area of 4. What is the circulation?
2 of 8
Curl is zero everywhere. Is the field conservative? 1 yes, 0 no.
3 of 8
A curl of 2 over a disc of radius 1. What is the circulation divided by π?
4 of 8
Green's theorem relates a boundary integral to what? 1 a point value, 2 an area integral.
5 of 8
A constant curl of 1 over an area of 12. What is the circulation?
6 of 8
Does a paddle wheel spin in a field with zero curl? 1 yes, 0 no.
7 of 8
Match each theorem to what it connects.
Tap a card on the left to start.
8 of 8
A region of area 0. What is the circulation around its boundary?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A constant curl of 4 over an area of 6. What is the circulation?
Question 2 of 2
What idea do all the big integral theorems share?
What You Learned
- Green's theorem equates the circulation around a closed curve with the curl integrated over the inside.
- Curl is the local spin of a field, the rate a tiny paddle wheel would turn.
- Every big theorem of this course says the same thing: the boundary knows the interior.