1. Draws a green closed curve through the seven points.

2. A closed curve bounding a plane area.

3. Well, you just take a closed curve in the plane.

4. If the interior of any closed curve in R -- - is also contained in R.

4. try its best to gather and create good sentences.

5. I need to be on a closed curve to do it.

6. If we have a closed curve then the line integral for work is just zero.

7. I know in advance that any closed curve, C so, C in particular, has to bound some surface.

8. That’s a puzzle: On a closed curve, the entropy has to finish exactly where it started, but the arrow of time says that entropy tends to increase and never decrease.

9. If I take any closed curve, the work will always be zero.

10. Stokes says if I have a closed curve in space, now I have to decide what kind of thing it bounds.

11. Green's theorem for flux says I have a closed curve that goes counterclockwise around some region.

12. If I am path independent, then if I take a closed curve, well, it has the same endpoints as just the curve that doesn't move at all.

13. You know it's automatically OK because if you have a closed curve, then the vector field is, I mean, if a vector field is defined on the curve it will also be defined inside.

14. The algebraic sum of the product of any point on a closed curve and the distance from this point's normal line to a fix point is zero.

15. And, I still want to compute the line integral along a closed curve.

16. To remind ourselves that we are doing it along a closed curve, very often we put just a circle for the integral to tell us this is a curve that closes on itself.

17. It says that the work done by a vector field along a closed curve can be replaced by a double integral of curl F.

18. It just reminds you that you are doing it on a closed curve.

19. A false target criterion is introduced to distinguish between potential false objects and true beacon which based on the closed curve fitting of image edge.

20. OK, so in particular, if you have a vector field that's defined everywhere the plane, then you take any closed curve.

21. Well, the line integral along C1 minus C2, well, C1-C2 let's just form a closed curve that is C1 minus C2.

22. It says that the line integral is zero along any closed curve.

23. It gives a relationship between a surface integral over an oriented surface and a line integral along a simple closed curve .

24. So, Green's theorem says that if I have a closed curve, then the line integral of F is equal to the double integral of curl on the region inside.