The document is a lecture on vector analysis that covers: 1. Coordinate systems including Cartesian, cylindrical, spherical, and their transformations. 2. Vectors including definitions, operations of addition and multiplication, dot and cross products. 3. Differential operators including gradient, divergence and curl and their physical meanings.

1. Listrik Magnet Umiatin, M.Si Sesion #01 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

2. Outline ILMU FISIKA PENGUKURAN BESARAN DAN SATUAN METODE ILMIAH 1/7/2011 © 2010 UniversitasNegeri Jakarta | www.unj.ac.id | 2

3. Review of Vector Analysis Lecture 1 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 3

4. 1. Coordinates 4 Cartesian 2-dim. 1 - Origin 0 2 - System of orthogonal axis (0xy) 3 - Unit vectors and y x 0 3-dim. Orientation of the three-dimensional system of coordinates: - screwrule - right hand rule z y x 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. 1. Coordinates Orientation rules 5 z z x x y y 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. 1. Coordinates Orientation rules 6 x y z x y z 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Polar (2-dim.)P(r,q,z) y z P(r,q) q q x x 0 q 0 P(r,q) and For both: and 3-dim.: 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. 1. Coordinates 8 Transformations 1 – From polar to cartesian x = r cosq y = r sinq z = z 1 – Fromcartesian to polar z r z y 0 q r x z = z 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. 1. Coordinates 9 Spherical 1 - Definitions z 2 - Transformations r sin j z j r y 0 q r sin j x 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. 1. Coordinates 10 Definition of radian B A q (for the disk : q= 2p radians) r r 0 From this definition: 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

14. Use a property of integralsx -r r 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. 1. Applications (1) 12 f(x) = px 1 – Triangle area from the equation pb = h If b is the basis and h the height: Equation: f(x)=px h 2 – Surface of a diskfrom the equation x 0 b x -r r 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. 1. Applications (1) 13 f(x) = px 1 – Triangle area from the equation pb = h If b is the basis and h the height: h 2 – Surface of a diskfrom the equation x 0 b Let x -r r 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. 1. Applications (2) 14 1 - Length of a circonference Contribution of the angle dq to the length: B A dq r r Total length: sum of contributions: 0 2 - Surface of a diskusing polar cordinates The contribution to the area of the sector having r as length and q as angle is the aerea of the triangle having r as basis and rdq as height: r dq r q r 0 dA= Total area : A= 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. 1. Applications (2) 15 1 - Length of a circonference Contribution of the angle dq to the length: B A dq r r Total length : 0 2 - Area of a diskusing polar cordinates The contribution to the area of the sector having r as length and q as angle is the aerea of the triangle having r as basis and rdq as height: r dq r q r 0 Total aerea : 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. 1. Applications (3) 16 1 – Surface of a triangle (base b , height: h) f(x)=px Contribution of the infinitesimal surface dy.dx : dA = pb=h h 2 – Surface of the ellipse x b 0 Equation: b dA= A= a 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

20. 1. Applications (3) 17 1 – Surface of a triangle (base b , height: h) f(x)=px Contribution of the infinitesimal surface dy.dx : dA = dy.dx pb=h Area: h 2 – Surface of the ellipse x b 0 b a 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

27. 2. Vectors 24 Geometric definition 1 - Modulus (length) > 0 : AB = 2 - Support (straight line): D, or every straight line parallel to D 3 - Direction (arrow) D B D’ A Consequence: if CD = AB if D’ // D and if the orientation is the same then: D C 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

28. 2. Vectors Algebric expression 25 y : components q x 0 q q 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

29. 2. Vectors 26 Definitions of operations on vectors 1 - Addition (Chasles relationship) B C The addition confers to the set of vectors a structure of commutative group ( is the neutralelement the opposite element) A 2 – Multiplication by a real number k Distributivity/addition: These 2 operations confer to the set of vectors a structure of commutative ring (k=1 is the neutral element) 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

30. 2. Vectors 27 Dot product 1 – Geometric definition (commutativity) H q 2 - Orthonormality relationship 0 3 – Algebric expression 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

32. 2. Vectors 29 Dot product: other notations is a matrix with one column and 3 rows: 1 - Matrix: 2 - Einstein convention: implicit sum on repeated indices 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

33. 2. Vectors 30 Using Einstein convention Total differential Product of matrices Trace of a matrix 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

34. 2. Vectors 31 Cross or vector product 1 – Geometric definition q N.B.: 2 – Properties: - anticommutativity: - bilinearity 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

35. 2. Vectors Cross product: other notations 32 3 – Algebric expression: 4 – Einstein convention: Levi-Civita symbol 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

37. 2. Vectors Cross or vector product: computation 34 Sarrus rule: 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

38. 2. Vectors Double cross product 35 (bac – cab or abacab rule) In order to apply the relationship between Levi-Civita and Kronecker symbols, both Levi-Civita symbols have to begin with the same indice k. Then we use the invariance by rotating indices. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

39. 2. Vectors Vector triple product 36 1 – Definition: 2 – Expression: 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

40. 2. Vectors 37 Properties of triple product Proof. Consider, for example, the first equation: Other proof of the first equation using Levi-Civita symbols: Wecan permute a and c (but not indices) in Levi-Civita symbol 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

41. 3. Differential operators 38 Gradient 1- Definition. f(x,y,z) is a differentiable scalar field 2 – Physical meaning: is the local variation of f along dr. Particularly, gradf is perpendicular to the line f = ctt. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

42. 3. Differential operators 39 Divergence 1 – Definition is a differentiable vector field x x+dx 2 – Physical meaning is associated to local conservation laws: for example, we’ll show that if the mass of fluid (or of charge) outcoming from a domain is equal to the mass entering, then is the fluid velocity (or the current) vectorfield 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

43. 3. Differential operators 40 Curl is a differentiable vector field 1 – Definition. 2 – Physicalmeaning: isrelatedto the local rotation of the vectorfield: If is the fluid velocity vectorfield 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

44. 3. Differential operators 41 Laplacian: definitions 1 – Scalar Laplacian. f(x,y,z) is a differentiable scalar field is a differentiable vector field 2 – VectorLaplacian. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

45. 3. Differential operators 42 Laplacian: physical meaning As a second derivative, the one-dimensional Laplacianoperatorisrelated to minima and maxima: when the second derivativeis positive (negative), the curvatureis concave (convexe). f(x) convex concave x In most of situations, the 2-dimensional Laplacianoperator is also related to local minima and maxima. If vE is positive: E 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

46. 3. Differential operators 43 Summary resp. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

50. 3. Differential operators 47 Conservative vectorfield then Theorem. If there exists f such that H P Consequently, the value of the integraldoesn’tdepend on the path, but only onitsbeginning A and its end B. Wesaythat the vectorfieldisconservative Proof. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

51. 3. Differential operators 48 1st Stokes formula: vectorfield global circulation Theorem. If S(C)is any oriented surfacedelimited by C: S(C) C Sketch of proof. y Vy . Vx . . x P . … and then extend to any surface delimited by C. 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

52. 3. Differential operators 49 2nd Stokes formula: global conservation laws Theorem. If V(C)is the volumedelimited by S Sketch of proof. Flow through the oriented elementary planes x = ctt and x+dx = ctt: x x+dx -Vx(x,y,z).dydz + Vx(x+dx,y,z).dydz and then extend this expression to the lateral surface of the cube. Other expression: extended to the vol. of the elementary cube: 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

53. 3. Differential operators 50 Vector identities Use Einstein convention and Levi-Civita symbol to show them curl(gradf) = 0 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

54. Thank You 1/7/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 51