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16.360 Lecture 13

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  1. 16.360 Lecture 13 Basic Laws of Vector Algebra Scalars: e.g. 2 gallons, $1,000, 35ºC Vectors: e.g. velocity: 35mph heading south 3N force toward center

  2. 16.360 Lecture 13 • Cartesian coordinate system z A  y  x

  3. 16.360 Lecture 13 • Vector addition and subtraction C = B+A = A +B, A C C parallelogram rule A head-to-tail rule B B |C| = |B+A| = |A| +|B|, D = A - B = -(B –A), A A D D’ = (B-A) B B

  4. 16.360 Lecture 13 • position and distance z A B y D = A - B = -(B –A), x

  5. 16.360 Lecture 13 • Vector multiplication 1. simple product 2. scalar product (dot product)

  6. 16.360 Lecture 13 Properties of scalar product (dot product) a) commutative property b) Distributitve property

  7. 16.360 Lecture 13 3. vector product (cross product) a) anticommutative property b) Distributitve property c)

  8. 16.360 Lecture 13 3. vector product (cross product)

  9. 16.360 Lecture 13 Example vectors and angles • In Cartesian coordinate, vector A is directed from origin to point P1(2,3,3), and vector B is directed from P1 to pint P2(1,-2,2). Find: • (a) Vector A, its magnitude |A|, and unit vector a • (b) the angle that A makes with the y-axis • (c) Vector B • (d) the angle between A and B • (e) perpendicular distance from origin to vector B

  10. 16.360 Lecture 13 4. Scalar and vector triple product a) scalar triple product b) vector triple product

  11. 16.360 Lecture 13 Example vector triple product

  12. 16.360 Lecture 14 • Cartesian coordinate system z dl   y x

  13. 16.360 Lecture 14 • Cartesian coordinate system z directions of area y x

  14. z y x 16.360 Lecture 14 • Cylindrical coordinate system

  15. z y x 16.360 Lecture 14 • the differential areas and volume

  16. 16.360 Lecture 14 Example: cylindrical area z 5 3 60 y 30 x

  17. 16.360 Lecture 14 • Spherical coordinate system

  18. 16.360 Lecture 14 • differential volume in Spherical coordinate system

  19. 16.360 Lecture 14 • Examples (1) Find the area of the strip (2) A sphere of radius 2cm contains a volume charge density Find the total charge contained in the sphere

  20. 16.360 Lecture 15 • Cartesian to cylindrical transformation

  21. 16.360 Lecture 15 • Cartesian to cylindrical transformation

  22. 16.360 Lecture 15 • Cartesian to cylindrical transformation

  23. 16.360 Lecture 15 • Cartesian to Spherical transformation

  24. 16.360 Lecture 15 • Cartesian to Spherical transformation

  25. 16.360 Lecture 15 • Cartesian to Spherical transformation

  26. 16.360 Lecture 15 • Distance between two points:

  27. 16.360 Lecture 16 Gradient in Cartesian Coordinates Gradient: differential change of a scalar The direction of is along the maximum increase of T.

  28. 16.360 Lecture 16 Example of Gradient in Cartesian Coordinates Find the directional derivative of along the direction and evaluate it at (1, -1,2).

  29. 16.360 Lecture 16 Gradient operator in cylindrical Coordinates

  30. z y x 16.360 Lecture 16 Gradient operator in cylindrical Coordinates

  31. 16.360 Lecture 16 Gradient operator in Spherical Coordinates

  32. 16.360 Lecture 16 Properties of the Gradient operator

  33. 16.360 Lecture 17 Flux in Cartesian Coordinates

  34. 16.360 Lecture 17 Flux in Cartesian Coordinates

  35. 16.360 Lecture 17 Definition of divergence in Cartesian Coordinates

  36. 16.360 Lecture 17 Properties of divergence If No net flux on any closed surface. Divergence theorem

  37. z y x 16.360 Lecture 17 Divergence in Cylindrical Coordinates

  38. z y x 16.360 Lecture 17 Divergence in Cylindrical Coordinates

  39. 16.360 Lecture 17 Divergence in Spherical Coordinates

  40. 16.360 Lecture 17 Divergence in Spherical Coordinates

  41. 16.360 Lecture 17 Divergence in Spherical Coordinates

  42. 16.360 Lecture 18 Circulation of a Vector

  43. 16.360 Lecture 18 Circulation of a Vector

  44. 16.360 Lecture 18 Curl in Cartesian Coordinates

  45. 16.360 Lecture 18 Vector identities involving the curl Stokes’s theorem

  46. 16.360 Lecture 18 Curls in Rectangular, Cylindrical and Spherical Coordinates

  47. 16.360 Lecture 18 Laplacian Operator of a scalar

  48. 16.360 Lecture 18 Laplacian Operator of a vector