Green's Theorem and What Comes Next

The boundary knows what the inside is doing.

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

NameDate

Figures — use these to answer the problems

This field has constant curl, so it circulates everywhere. Walk any loop and the circulation is proportional to the area you enclosed.

the amount of spin1

start x2

start y0

Figure 1

A shear field looks like it is going nowhere but has curl all the same. A paddle wheel dropped here still turns.

the shear rate1

Figure 2

Straightforward practice. Get the method working first.

5 problems

A constant curl of 4 over an area of 6. What is the circulation?
Answer
2
What idea do all the big integral theorems share?
- What happens on the boundary is decided by what happens inside.
- Every integral eventually becomes zero.
A constant curl of 5 over an area of 4. What is the circulation?
Answer
Curl is zero everywhere. Is the field conservative? 1 yes, 0 no.
Answer
A curl of 2 over a disc of radius 1. What is the circulation divided by π?
Answer

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
A constant curl of 1 over an area of 12. What is the circulation?
Answer
Does a paddle wheel spin in a field with zero curl? 1 yes, 0 no.
Answer

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

4 problems

9
Match each theorem to what it connects.
Draw a line from each item on the left to its match on the right.
- The Fundamental Theorem of Calculus
- Green's theorem
- The Divergence theorem
- Two endpoints to an interval
- A closed curve to the region inside it
- A closed surface to the solid inside it
A region of area 0. What is the circulation around its boundary?
Answer
The Circulation
A constant curl of 3 over a region of area 10. What is the circulation around the boundary?
Answer
The Still Field
A field with curl zero everywhere. What is the circulation around any closed loop?
Answer