1. EMLAB 1 Chapter 1. Vector analysis

2. EMLAB 2 Mathematics -Glossary Scalar : a quantity defined by one number (eg. Temperature, mass, density, voltage,... ) Vector : a quantity defined by a set of numbers. It can be represented by a magnitude and a direction. (velocity, acceleration, …) Field : a scalar or vector as a function of a position in the space. (scalar field, vector field, …) Scalar field Vector field

3. EMLAB 3 Scalar fieldVector field Examples Thermal image Air flow

4. EMLAB 4 Vector algebra Unit vector : ‘^’ means the unit vector. Components of a vector Commutative law Associative law

5. EMLAB 5 Products of vectors scalar product : the result of a scalar product is a scalar.

6. EMLAB 6 The magnitude of C is the area of a parallelogram made by two vectors A and B. Products of vectors vector product : : the result of a vector product is a vector which is normal to the original vectors.

7. EMLAB 7 Calculation of vector products (+) (-) (+)

8. EMLAB 8 (+) (-) If rotated in the direction of x,y,z, the sign of the result is (+). If rotated in the opposite direction, the sign is (-). The same rule applied to other coordinate systems with change of unit vectors. Cyclic relations of unit vectors

9. EMLAB 9 Example of a scalar product

10. EMLAB 10 Example of vector product torque Angular acceleration is proportional to the applied torque. Torque is proportional to the product of radius and force. If the sum of torques due to A and B has nonzero value, the seesaw is rotated.

11. EMLAB 11 Vector identity The order of operators and can be interchanged. The scalar triple product is equal to the volume of the parallelogram defined by the three vectors. Scalar triple product The result is the same despite the change of the order of and x interchanged.

12. EMLAB 12 Quantities represented by vectors Position vector : a position can be represented by a vector whose origin is specified by O. Field vector : A vector that represents physical quantities at the position specified by a position vector. Position vector Field vector Origin

13. EMLAB 13 O Origin Position vector Example : Velocity Velocity vector

14. EMLAB 14 Coordinate system Position vector : A position can be specified with a vector whose origin is at a point O. O Observation position is written by R. Source position is represented by a primed vector. 1.A coordinate system is only an apparatus that describes positions and physical phenomena. 2.Physical laws are independent of coordinate systems adopted by any observers.

15. EMLAB 15 Field vectors

16. EMLAB 16 We can choose a set of unit vectors which fits a specific problem. Some orthogonal coordinate systems and their unit vectors constant unit vectors The directions of two unit vectors change with observation positions.

17. EMLAB 17 O O O Position vectors

18. EMLAB 18 For cylindrical coordinate systems, unit vectors ρ and φ are functions of an observer’s position.

19. EMLAB 19 For spherical coordinate systems, unit vectors r, θ, and φ are functions of a position.

20. EMLAB 20 Circular cylindrical coordinate Cylindrical coordinate

21. EMLAB 21 Example D1.5. Represent the position D(x= -3.1, y= 2.6, z= -3) using circular cylindrical coordinate system.

22. EMLAB 22 Example For an observation point P(1,2,3) in rectangular coordinate, express the position vector and E-field vector using unit vectors of cylindrical coordinate. (1) Position vector (2) Field vector

23. EMLAB 23 Spherical coordinate

24. EMLAB 24 (+) (-) (+) (-) (+) (-) If rotated in the direction of x,y,z, the sign of the result is (+). If rotated in the opposite direction, the sign is (-). The same rule applied to other coordinate systems with change of unit vectors. Cyclic relations of unit vectors

25. EMLAB 25 Example For an observation point P(1,2,3) in rectangular coordinate, express the position vector and E-field vector using unit vectors of spherical coordinate. (1) Position vector (2) Field vector

26. EMLAB 26 O Origin Position vector changed Displacement vector Instantaneous change of position vector

27. EMLAB 27 Rectangular coordinate If the position can be described by line segments, rectangular coordinate systems are convenient. Unit vectors are parallel to x, y, z axes. Their directions are fixed. (The unit vectors are constant ones.) Position vector changed Displacement vector O Origin

28. EMLAB 28 Cylindrical coordinate The direction of ρ is defined as that points away from z-axis. Φ-direction is orthogonal to ρ.

29. EMLAB 29 Unit circle : the magnitude of unit vectors are unity.

30. EMLAB 30 Spherical coordinate R : points away from the origin. θ : longitudinal direction Φ: latitude direction

31. EMLAB 31 Unit circle : the magnitude of unit vectors are unity.

32. EMLAB 32 Multiple integrals Line integral

33. EMLAB 33 Surface integral dx dy n : a unit vector normal to the integration surface Surface integrals

34. EMLAB 34 To represent arbitrary physical vector quantities, the number of unit vectors in a rectangular coordinate is three. The unit vectors are parallel to x, y, z axes and the directions are constant. Rectangular coordinate Line integral Surface integral Volume integral

35. EMLAB 35 Line integral Example

36. EMLAB 36 Cylindrical coordinate 1.The unit vectors for a cylindrical coordinate are rho, phi, z. 2.The unit vector z is constant vector. 3.The directions of rho and phi are changing with positions. Line integral Surface integral Volume integral

37. EMLAB 37 Example Evaluate a line integral which has an integrand of F∙dr.

38. EMLAB 38 Spherical coordinate Unit vectors for a spherical coordinate are r, theta, phi. The directions of them all change with positions. Line integral Surface integral Volume integral

39. EMLAB 39 Example θ : longitudinal direction

40. EMLAB 40 Summary The unit vectors for a rectangular coordinate system are constant. For spherical or cylindrical coordinates, the direction of unit vectors change with position. Cylindrical coordinate Spherical coordinate Rectangular coordinate

41. EMLAB 41 Coordinate transformation Cylindrical coordinate Coordinate transformation matrix A is an orthogonal matrix. If a matrix is orthogonal, A -1 is equal to A T. Spherical coordinate

42. EMLAB 42 Integral of vector function