MATH 250: Calculus III

May 12, 2012

Final Exam Review Topics

1. Parametric equations of curves in the plane; paths versus curves

2. Slope of a tangent line to a curve given in parametric form

3. Arc length of a curve given in parametric form

4. Speed along a parametrized path

5. Polar coordinates; conversions between polar and rectangular coordinates; graphing curves and regions in polar coordinates

6. Arc lengths in polar coordinates

7. Areas in polar coordinates; area between two curves in polar coordinates

8. Conic sections: properties; standard forms; foci, directrix, etc.

9. Vectors in two dimensions; magnitude of a vector; vector algebra; basic properties; triangle inequality

10. Unit vectors, standard basis vectors

11. Vectors in three dimensions; distance formula

12. Parametric equations of a straight line in three dimensions; intersections of straight lines

13. Dot product; basic properties (with proofs); angle between vectors; projection of a vector along another vector

14. Determinants

15. Cross product; basic properties (with proofs)

16. Areas and volumes with the use of cross product

17. Planes in three dimensions; planes through three given points; intersection of a plane and a line; angle between planes; traces of a plane

18. Basic quadric surfaces; traces of quadric surfaces

19. Cylindrical coordinates; conversions between cylindrical and rectangular coordinates; level surfaces; identifying surfaces and regions in cylindrical coordinates

20. Spherical coordinates; conversions between spherical and rectangular coordinates; level surfaces; identifying surfaces and regions in spherical coordinates

21. Vector-valued functions: parametric curves in three dimensions; projections of curves onto coordinate planes

22. Calculus of vector-valued functions

23. Arc length; speed

24. Arc length parametrization

25. Unit tangent vector; curvature; unit normal vector

26. Motion in three-dimensional space; velocity and acceleration; finding velocity and path from acceleration

27. Tangential and normal components of acceleration

28. Functions of two variables; traces; level curves and contour maps

29. Functions of three variables; level surfaces

30. Limits and continuity of functions of several variables

31. Partial derivatives; higher-order partial derivatives; Clairaut’s theorem

32. Linearizations; linear approximations; tangent planes to surfaces z=f(x,y)

33. Gradient

34. Chain rule for paths

35. Directional derivatives

36. Significance of gradient: direction and value of maximum rate of increase; normal vector to level surface; finding tangent planes to level surfaces

37. Chain rule for functions of several variables

38. Critical points; local extrema

39. Global extrema

40. Definition of double integral; double integral as volume

41. Iterated integrals; Fubini’s theorem

42. Double integrals over regions between curves

43. Interchanging the order of double integration

44. Derivation and definition of a triple integral

45. Iterated integrals and Fubini’s theorem for triple integrals

46. Triple integrals over general regions

47. Triple integrals over regions between surfaces

48. Changing the order of triple integration

49. Computing volumes with double and triple integration

50. Double integrals in polar coordinates

51. Triple integrals in cylindrical and spherical coordinates

52. Change of variables; jacobian

53. Vector fields in two and three dimensions

54. Line integrals of scalar fields; derivation and computation

55. Line integrals of vector fields; derivation and computation

56. Line integral as work

57. Conservative vector fields; gradient of potential function

58. Finding potential functions in two and three dimensions

59. Fundamental theorem for gradient vector fields; path independence

60. Circulation of a vector field

61. Theorems about conservative vector fields

62. Rotation (curl) of a vector field

63. Surface parametrization

64. Normal vectors of parametric surfaces; tangent planes

65. Surface area of parametric surfaces

66. Surface integrals of scalar functions

67. Surface integrals of vector fields

68. Flux of a vector field

69. Scalar curl (curl in 2D)

70. Green’s Theorem

71. Computing area via Green’s Theorem

72. Stokes’ Theorem

73. Divergence of a vector field

74. Divergence Theorem

75. The gradient operator unifying notation

76. The Big Picture of Calculus