Notes for Analysis Et/Wi

1. Notes for Analysis Et/Wi Third Quarter GS TU Delft 2001 - 2002

2. Week 1. Surfaces and Tangential Lines

3. Week 1. Surfaces and Tangential Planes

4. Week 1. Parameter notationfor Tangential Lines and Planes

5. Week 1. Example: Tangent Lines but no Tangent Plane

6. Week 1. Reminder:Differentiable in 1-d Theorem: tangent line exists  differentiable

7. Week 1. Differentiable in 2-d Definition

8. Week 1. Linearization Definition (reminder) The linearization in a gives the tangent line to f in a. Definition The linearization in (a,b) gives the tangent plane to f in (a,b).

9. Week 1. Differentiable and Partially Differentiable Theorem

10. Week 1. Partially Differentiableand Differentiable Theorem (usually quite unworkable)

11. Week 1. Chain Rule Theorem: the chain rule in 2-d

12. Week 1. Example, Implicit function, a This approach is justified by the Implicit Function Theorem, which is skipped in the present course.

13. Week 1. Example, Implicit function, b

14. Week 1. Example, Implicit function, c

15. Week 2. Tangent lines in other directions

16. Week 2. Unit vector Definition

17. Week 2. Directional derivative, the definition Definition

18. Week 2. Differentiable, three flavours Theorem

19. Week 2. The gradient in 2 dimensions Definition Theorem And a similar definition and theorem for 3 dimensions ….

20. Week 2. Tangent planes to level surfaces

21. Week 2. The gradient and the steepest ascent/descent

22. Week 2. The gradient field

23. Week 2. Definition ofmaxima and minima Definition Definition And similar definitions for local and absolute minimum.

24. Week 2. Maxima and minima

25. Week 2. Pure 2nd order functions a, a maximum a minimum

26. Week 2. Pure 2nd order functions b, many minima a saddle

27. Week 2. Test for maximum, minimum or saddle

28. Week 2. Second derivatives test Theorem (the second derivatives test)

29. Week 2. Open and closed Definitions Open, closed and neither, but all 3 are bounded

30. Week 2. Continuous function on closed bounded set Theorem

31. Week 3. Integrals in two dimensions,rectangular domains I

32. Week 3. Integrals in two dimensions,rectangular domains II

33. Week 3. Integrals in two dimensions,rectangular domains III

34. Week 3. Approximation by stepfunctions And we don’t care how it is defined on the edges.

35. Week 3. The integral of a stepfunction

36. Week 3. Defining the integral as a limit through stepfunctions I Definition of Riemann-integrable In simple words: there exist stepfunctions above and below the function which have integrals that are arbitrary close.

37. Week 3. Defining the integral as a limit through stepfunctions II Definition of Riemann-integral

38. Week 3. Continuous functions, Riemann-sums and integrability Theorem: Continuous functions on a rectangle are Riemann-integrable and the Riemann-sums converge to the integral. a Riemann-sum

39. Week 3. Properties of the integral From now on we skip Riemann and will just say integrable. Just like the one-dimensional integral ….

40. Week 3. The average of a function Definition:

41. Week 3. From 2-d integral to iterated 1-d integral Fubini’s Theorem This result allows one to compute a 2-d integral.

42. Week 3. Integrals over general domains

43. Week 3. Type I and II domains

44. Week 3. Other domains

45. Week 3. Integrals over general domains, an example

46. Week 3. Integrals over general domains:‘the proof of the pudding is in the eating’.

47. Week 4. Double integrals in polar coordinates

48. Week 4. Double integrals in polar coordinates, heuristics Cartesian coordinates Polar coordinates

49. Week 4. Double integrals to iteratedintegrals by polar coordinates Fubini’s Theorem in polar coordinates Sometimes such a domain is called a polar rectangle.

50. Week 4. Integrating with polar coordinates, an example