Three Gentlemen and the Amoebas

1. Three Gentlemen and the Amoebas - a drama in three acts

2. Act I. George Green (1793-1841) • Born July 14, 1793, in Nottingham, England. A self-taught mathematician, the only son of a semi-literate baker, he spent only a year at age 8 in an academy in Nottingham, and then his father set him to work in the bakery. • In 1807 the baker built a windmill, and at age 14 George started an apprenticeship under his father's mill manager, William Smith, who had a daughter, Jane Smith, with whom he had seven children but never married . • George studied mathematics on the top floor of the mill, entirely on his own. In 1828, at age 35, he published one of the most important mathematical works of all time: An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, which was sold 51 copies.

4. Green took Sir Edward Bromhead's advice, left his mill and became an undergraduate at Cambridge in October 1833 at the age of 40. • His academic career was excellent, and after his graduation in 1837 he stayed on the faculty at Gonville and Caius College. He wrote on optics, acoustics, and hydrodynamics. None of them is as important as his 1828 essay. In 1840 he became ill and returned to Nottingham, where he died the next year. • Green's work was not well-known in the mathematical community during his lifetime. In 1846, Green's work was rediscovered by Lord Kelvin, who popularized it for future mathematicians

6. Recall: Gradient Vector Field Thus, define formally the Del Operator:

7. Can we apply the Del operator  to vector-Valued functions? Answer: Yes. In two different ways: F is called the divergence of F and can be written as “div F”. F is called the curl of F and can be written as “curl F”.

8. Divergence (Dot Product) • F , a scalar valued function, is called the divergence of F and can be written as “div F”.

9. Note that the divergence of a vector field F is a scalar function.

10. Meaning of Divergence It measures the net inflow or outflow of a vector field at a point.

11. F div Frepresents the rate of expansion per unit volume under the flow F of a gas or fluid.

12. Divergence div F > 0 div F = 0 div F < 0

13. The vector field F = x i

14. The vector field F = x i +y j

15. The vector field F = -x i + -y j

16. The vector field F = -y i +x j

17. The vector field F = x i +(-y)j

18. The Curl (Cross Product) • F , a vector field, is called the curl of F and can be written as “curl F”.

20. Meaning of Curl The curl of the wind vector field F, curl F, measures its spinning effect. It is a vector field that lines up with the axis along which the wind is trying to twirl you, and whose magnitude indicates the strength of the twirling effect, in the counter-clockwise direction.

21. The velocity vector vand the angular velocity ωof a rotating body are related by . If ω= ωk is directed along z axis, then v = -ωyi + ωxj and we obtain

22. Meaning of the curl Right-Hand Principle

23. The vector field F = -y i +x j Curl F = 2 k

24. F(x, y) = The unit vector field has curl F =

25. Vector Field F = y i – x j Curl F = -2 k

26. The Vector Field v(x, y) = (y i – x j) / (x2+y2) Curl v = 0 !!

27. Looking at a paddle wheel from above a moving fluid. The velocity field has curl v = 0, and hence is irrotational; i.e., the paddle wheel does not rotate around its axis ω . Curl v = 0

28. the unit vector field, has curl u = But notice that for

31. Scalar Curl • If F = P(x,y)i + Q(x,y)j is a vector field on the plane, regarded as a vector field in R3 for which the k component is zero, then Curl F = k . • The scalar function of (x, y) is called the scalar curl of F.

32. Green’s Theorem • If c is a piecewise-smooth simple closed curve oriented counterclockwise, D is the region bounded by c, and P and Q are differentiable on and around D, then D c

33. Proof Idea of Green’s Theorem • Split the double integral into two. • Integrate Q/x with respect to x first and P/y with respect to y first. • The reason for the “” sign is that, when traveling counterclockwise on c, dx is negative on the upper part.

34. Green’s Theorem

36. Proof. For y-simple regions,

37. For x-simple regions, we have similarly QED

38. Green’s theorem generalizes to regions which can be broken into pieces, each of which is simple. In particular, it holds for closed bounded region in R2 whose boundary consists of finitely many smooth simple closed curves

39. Green’s Theorem for the above mentioned region; where is the boundary of the region with positive orientation.

40. Application of Green’s Theorem Proof. Let P(x, y) = -y and Q(x, y) = x. Then by Green’s Theorem,

41. ds ds = c’(t) dt curl F · k is just the scalar curl.

42. Meaning of Curl on the Plane • If F is the velocity vector, then • The above theorem shows that the total (net) “twirling” of the vector field F over the region D is equal to the circulation of F along its boundary ∂D. is the total circulation of the flow along ∂D .

43. In , D can be arbitrarily small. Hence around any point p = (x1, y1) in D, we can choose a small disk E and let it shrink to p : Meaning of Curl on the plane

44. c’(t) = (x’(t), y’(t)) the outward unit normal vector

45. Proof. • c’(t) = (x’(t), y’(t)) is tangent to ∂D and n•c’ = 0. • Let F = (P, Q). By the definition of line integral • By Green’s theorem, QED

46. Physical Meaning of Divergence on the Plane • If F is the velocity vector, then is the net quantity of fluid to flow across ∂D per unit time; that is, the rate of fluid flow, also called the flux across ∂D. • The divergence theorem shows that the total (net) divergence of the vector field F over the region D is equal to the flux of F across its boundary ∂D.

47. Meaning of Divergence • Using mean-value theorem for double integral, it follows from the divergence theorem that for a small disk E around a fixed point P = • Hence div F is the rate of net outward flux per unit area at the point P= .

48. The Party Line Integral* A huge rambunctious party is going on inside a large room. Numerous people mill about, coming and going. The motion of the party goers forms a party velocity vector field. (Mathematicians analyze these fields by studying the patterns of drool and spilled beer left on the floor.) Certain people act like magnets. Other party goers are attracted to them, sometimes because of their scintillating conversation, and sometimes because of how they look in a sleeveless T‑shirt. They are surrounded by admirers. Though they don't often think of themselves that way, they are what we call sinks. Other party goers are repellers, sometimes because they can only talk about their most recent root canal and sometimes because of how they look in a sleeveless T‑shirt. Revelers move away from them as quickly as politeness allows, and often quicker. They are known as sources. Most people are neither sources nor sinks, but just go with the flow.

49. The divergence theorem tells us the relationship between the sources and sinks and the number of people entering and leaving the room. It says that if we sum up, by integrating, the divergence of the party vector field, then we get the total number of people entering or leaving the room, either by sneaking past the bouncers at the door or by falling off the balcony into the pool. *How to Ace the rest of Calculus: The Streetwise Guide 2001 and Jerry Kazdan.