Green's Theorem and What Comes Next

The boundary knows what the inside is doing.

12 problems · about 23 minutes · N-VM.C.11

What this lesson teaches

The student states Green's theorem and recognises it as a form of the Fundamental Theorem of Calculus.

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.

Straightforward practice. Get the method working first.

5 problems

A constant curl of 4 over an area of 6. What is the circulation?
Answer 24
Why 24.
What idea do all the big integral theorems share?
Answer What happens on the boundary is decided by what happens inside.
Why Boundary and interior determine each other.
A constant curl of 5 over an area of 4. What is the circulation?
Answer 20
Why Multiply.
Curl is zero everywhere. Is the field conservative? 1 yes, 0 no.
Answer 1
Why No spin means path independence.
A curl of 2 over a disc of radius 1. What is the circulation divided by π?
Answer 2
Why Area π times curl 2.

The same ideas with more to keep track of.

3 problems

Green's theorem relates a boundary integral to what? 1 a point value, 2 an area integral.
Answer 2
Why The interior of the curve.
A constant curl of 1 over an area of 12. What is the circulation?
Answer 12
Why The circulation equals the area.
Does a paddle wheel spin in a field with zero curl? 1 yes, 0 no.
Answer 0
Why Curl is exactly that spin.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each theorem to what it connects.
Answer The Fundamental Theorem of Calculus → Two endpoints to an interval; Green's theorem → A closed curve to the region inside it; The Divergence theorem → A closed surface to the solid inside it
Why Each one raises the dimension by one.
A region of area 0. What is the circulation around its boundary?
Answer 0
Why Nothing is enclosed.
The Circulation: A constant curl of 3 over a region of area 10. What is the circulation around the boundary?
Answer 30
Why 30.
The Still Field: A field with curl zero everywhere. What is the circulation around any closed loop?
Answer 0
Why 0.