I Think, Therefore I Exist

Chapter 3 Part 2: Coordinate Systems

I think, therefore I exist.

 Introduction . Orthogonal coordinate systems: Axes are mutually perpendicular (dot products are zero). . Nonorthogonal coordinate systems are seldom used. . A complex problem can be greatly simplified when it is formulated in an appropriate coordinate system because of symmetry (One or more variables are constant). . Common coordinate systems: o Rectangular (Cartesian) o Cylindrical o Spherical  Cartesian coordinate system (x, y, z) . It was introduced by the famous 16th century French philosopher Rene Descartes (Cartesian). . The axes are independent of the vector being measured.

. Differential length: dl  xˆ dx  yˆ dy  zˆ dz . Differential surface:

ds x  xˆ dy dz

ds y  yˆ dxdz

ds z  zˆ dxdy . Differential volume: dv  dxdy dz  Cylindrical coordinate system (r, , z) . r – Radial distance in the x-y plane (0 , )  – Azimuth angle measured from the x-axis (0 , 2) z – z-axis (- , ) . Only the z-axis is independent of the vector being measured.

. Orthogonal relationship: rˆ ˆ  zˆ ˆ zˆ  rˆ zˆ rˆ  ˆ rˆ rˆ 1 ˆˆ 1 zˆ zˆ 1 rˆ rˆ  0 ˆˆ  0 zˆ zˆ  0

. Differential elements: dlr = dr, dl = r d, dlz = dz. . Differential length: dl  rˆ dr  ˆ r d  zˆ dz . Differential surface:

dsr  rˆ r d dz ˆ ds   dr dz

ds z  zˆ r d dr . Differential volume: dv  r dr d dz  Spherical coordinate system (R, , ) . R – Radial distance from the origin (0 , )  – Zenith angle measured from the z-axis (0 , )  – Azimuth angle measured from the x-axis (0 , 2) . The axes are dependent of the vector being measured. . Orthogonal relationship: Rˆ ˆ  ˆ ˆ ˆ  Rˆ ˆ  Rˆ  ˆ Rˆ  Rˆ  1 ˆ ˆ  1 ˆ ˆ  1 Rˆ  Rˆ  0 ˆ ˆ  0 ˆ ˆ  0 . Differential elements: . dlR = dR,

. dl = R d,

. dl = R sin  d ˆ ˆ ˆ . Differential length: dl  R dlR  R d  Rsin dl . Differential surface: ˆ 2 ds R  R R sin d d ˆ ds  Rsin dR d ˆ ds   R dRd . Differential volume: dv  R2 sin dR d d  Coordinate system transformation: Caution – The sign of tan-1:  = tan-1(b/a) if a > 0  = 180 + tan-1(b/a) if a < 0 MatLab commands: . Rectangular to spherical: [th,phi,R]=cart2sph(x,y,z) EDU» [th,phi,R]=cart2sph(1,2,3) th = 1.1071 phi = 0.9303 R = 3.7417 . Spherical to rectangular: [x,y,z]=sph2cart(th,phi,R) EDU» [x,y,z]=sph2cart(1,2,3) x = -0.6745 y = -1.0505 z = 2.7279 . Rectangular to cylindrical: [phi,r,z]=cart2pol(x,y,z) EDU» [phi,r,z]=cart2pol(1,2,3) phi = 1.1071 r = 2.2361 z = 3 . Cylindrical to rectangular: [x,y,z]=pol2cart(th,r,z) EDU» [x,y,z]=pol2cart(1,2,3) x = 1.0806 y = 1.6829 z = 3  Distance between two points: 2 2 2 . Rectangular: d  x2  x1   y2  y1   z2  z1  2 2 2 . Cylindrical: d  r2  r1  2r2r1 cos(2  1)  z2  z1  2 2 . Spherical: d  R2  R1  2R2 R1cos 2 cos1  sin 2 sin1 cos(2 1 )

. Example:x1  1 y1  2 z1  3 x2  4 y2  5 z2  6

2 2 2 2 r1  x1  y1 r1  2.236 r2  x2  y2 r2  6.403

 y1   y2  1  atan  1  63.435 deg 2  atan  2  51.34 deg  x1   x2  2 2 2 2 2 2 R1  x1  y1  z1 R1  3.742 R2  x2  y2  z2 R2  8.775

 2 2   2 2   x1  y1   x2  y2  1  atan  1  36.699 deg 2  atan  2  46.862 deg  z1   z2 

2 2 2 x2  x1  y2  y1  z2  z1  5.196

2 2 2 r2  r1  2r1r2cos2  1  z2  z1  5.196

2 2 R2  R1  2R1R2cos2cos1  sin1sin2cos2  1  5.196