MODULE 2

• Rectangular Coordinate System


  A point is represented as (x, y) for 2D or (x, y, z) for 3D

  
• Cylindrical Coordinate System


  A point is represented as (ρ, φ, z)

  
• Spherical Coordinate System


  A point is represented as (r, θ, φ)

  
• Constant-Coordinate Surfaces


  Surfaces where one coordinate is constant
• Coordinate System


  A system composed of numbers, or coordinates, to determine the position of a point in space

  
• Types of Coordinate System


  • Orthogonal (perpendicular axes)
  • Non-orthogonal (non-perpendicular, oblique, skewed axes)

  
• Orthogonal Coordinate System


  Examples: Cartesian/Rectangular, Cylindrical, Spherical, Parabolic, Elliptical, and others

  
• Why use different coordinate systems


  • Ease of representation and computation
• Coordinate systems used in this course


  • Cartesian (Rectangular)
  • Cylindrical
  • Spherical

  
• Representing vectors in RCS then in CCS or CSC

  Using relationships between points and unit vectors
• Rectangular Coordinate System (RCS)


  Also known as Cartesian coordinate system. A point is represented as (x, y, z)

  
• Coordinate ranges in RCS: -∞ < x < ∞, -∞ < y < ∞, -∞ < z < ∞

  
• Vector in RCS


  Ā = Axax + Ayay + Azaz
• Magnitude of vector in RCS: |Ā| = √(Ax^2 + Ay^2 + Az^2)
• Point


  Defined with three values, referring to its intersection with the axes
• Line


  Defined with two values
• Plane


  Defined with a single value
• The intersection of two planes is a line. The intersection of all three planes is a point.
• Differential Surface Elements in RCS


  Top and Bottom: dx dy, Left and Right: dx dz, Front and Back: dy dz
• Differential Volume Element in RCS


  dx dy dz
• Cylindrical Coordinate System (CCS)


  A point is represented as (ρ, φ, z)

  
• Coordinate ranges in CCS: 0 ≤ ρ < ∞, 0 ≤ φ < 2π, -∞ < z < ∞

  
• Vector in CCS


  Ā = Aρaρ + Aφaφ + Azaz
• Magnitude of vector in CCS: |Ā| = √(Aρ^2 + Aφ^2 + Az^2)
• Differential Surface Elements in CCS

  ρ dφ dz, ρ dρ dz, dρ dφ
• Differential Volume Element in CCS

  ρ dρ dφ dz
• Dot Product in CCS

  Dot product of same unit vectors is 1, dot product of orthogonal unit vectors is 0
• Cross Product in CCS


  Each unit vector is perpendicular to each other because CCS is orthogonal
• Deriving dot products of RCS and CCS unit vectors

  Using vector projections and the azimuthal angle

  
• Finding rectangular unit vectors expressed in cylindrical unit vectors


  And finding cylindrical unit vectors expressed in rectangular unit vectors
• The colatitude and azimuthal angle must be given to be able to convert unit vectors from CCS to RCS
• How to get a vector from RCS to CCS


  1. Project vector A into the unit vectors 𝒂𝝆, 𝒂𝝓, 𝒂𝒏𝒅 𝒂𝒛

  2. Simplify to get the scalar components of the vector in CCS
• How to get a vector from CCS to RCS


  1. Project vector A into the unit vectors 𝒂𝒙, 𝒂𝒚, 𝒂𝒏𝒅 𝒂𝒛

  2. Simplify to get the scalar components of the vector in RCS
• Spherical Coordinate System (SCS)


  • Convenient when dealing with problems having spherical symmetry
  • A point in SCS is represented as (𝑟, 𝜃, 𝜙) since 3D
  • Coordinate ranges: 0 ≤ 𝑟 < ∞, 0 ≤ 𝜃 < ��, 0 < 𝜙 < 2𝜋

  
• Vector A in SCS

  • 𝑨 = 𝑨��𝒂𝒓 + 𝑨𝜽𝒂𝜽 + 𝑨𝝓𝒂𝝓
  • �� = √𝑨�� 𝟐 + 𝑨𝜽 𝟐 + 𝑨𝝓 𝟐
• Differential surface and volume elements for spherical coordinate system

  
• Dot product of unit vectors in SCS


  • Dot product of same unit vectors is equal to 1
  • Dot product of orthogonal unit vectors is equal to 0
• Cross product of unit vectors in SCS

  Each unit vector is perpendicular to each other because the spherical coordinate system is orthogonal

  
• Derivations of dot products of RCS and SCS unit vectors are done using vector projections
• Derivations are done using vector projections