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a. Coordinate System

Coordinate system for Cartesian, Cylindrical and Spherical

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a. Coordinate System

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  1. CARTESIAN, CYLINDRICAL AND SPHERICAL COORDINATE SYSTEM

  2. Coordinate System • oordinatSys • Coordinatessystemsareoftenusedtospecifythepositionofa point,buttheymayalsobeused tospecifythepositionofmore complexfiguressuchaslines,planes,circlesorspheres. • Thechoiceofthecoordinatesystemisbasedontheproblem oneisstudying. • Certainproblemsaresolvedeasilybyusingrectangular coordinatesystemswhereascertainothersarenot. • Somecoordinatesystemsmakemoresense,makeiteasierto describeasystem. • Coordinatesgiveyouasystematicwayofnamingthe pointsina space.

  3. 1. CARTESIANCOORDINATES, (x,y,z) • Intwodimensions,wecanspecifyapointonaplaneusingtwoscalar values,generallycalledxandy. • Wecanextendthistothree- dimensions,byaddingathird scalarvaluez. z-axis y-axis X P(x,y) P(x,y,z) Origin Y y-axis x-axis -∞ < x < ∞ -∞ < y < ∞ -∞ < z < ∞ P’ (x, y, z) Origin x-axis

  4. Note the coordinate values in the Cartesian system effectively representthedistancefromaplaneintersectingtheorigin. A vector in Cartesian coordinates can be written as (Ax, Ay, Az) or Ax ax+ Ay ay + Az az., Example , (2, -3, 4) or 2ax - 3ay + 4az The magnitude of A is

  5. if x = 0, it indicates 2-dimensional y - z plane. if y = 0, it indicates 2-dimensional x - z plane. If z = 0, it indicates 2-dimensional x - y plane.

  6. 2. CYLINDRICALCOORDINATES, (, , z) A point P in cylindrical coordinates is represented as (, , z) where. •  : the radius of the cylinder; radial displacement from the z-axis • : azimuthal angle or the angular displacement from x-axis • z : vertical displacement z from the origin (as in the Cartesian system). • The ranges of the coordinate variables ,  and z are • A vector A in Cylindrical coordinates can be written as • The magnitude of A is

  7. Cylindrical coordinate at point P(3,60,4)

  8. Cylindrical Coordinates

  9. 3. SPHERICALCOORDINATES (r, , , ) Point P represented as (r,θ,φ), where, • r : the distance from the origin, • θ : called the colatitude is the angle between z-axis and vector of P, • Φ : azimuthal angle or the angular displacement from x-axis (the same azimuthal angle in cylindrical coordinates). • The ranges of the coordinate variables r,  and  are • A vector A in Spherical coordinates can be written as • The magnitude of A is

  10. Spherical coordinate at point P(3,45,60)

  11. 4. COORDINATE TRANSFORMATION Sometimes, it is necessary to transform points and vectors from one coordinate system to another. Combiningtheresultsofthetwotrianglesallowsustowrite eachcoordinatesetintermsofeachother (i) CARTESIAN COORDINATE TO CYLINDRICAL COORDINATE (VICE VERSA) The transformation between these two coordinate systems are easily obtained from figure below. (a) Cartesian to cylindrical (b) Cylindrical to Cartesian

  12. (ii) CARTESIAN COORDINATE TO SPHERICAL COORDINATE (VICE VERSA) (c) Cartesian to spherical (d) Spherical to cartesian

  13. (iii) CYLINDRICAL COORDINATE TO SPHERICAL COORDINATE (VICE VERSA) (e) Spherical to cylindrical  (f) Cylindrical to spherical 

  14. (a) Cartesian to cylindrical Example 1 Express point P(3,4,5) from Cartesian to cylindrical Solution So, a point P (3, 4, 5) in Cartesian coordinate is the same as P ( 5, 0.927,5) in cylindrical coordinate

  15. Example 2 (c) Cartesian to spherical Express point P(-2,6,3) in spherical coordinate Solution =7  P(-2,6,3) = P(7,1.127 rad,1.89 rad)

  16. Example 3 (b) Cylindrical to Cartesian Convert (10, , -4) from Cylindrical to Cartesian. Solution  P (10, , -4) = P(5,8.77,-4)

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