COORDINATE SYSTEM & TRANSFORMATION,Transformation Matrix, DIFFERENTIAL VECTOR,Curvilinear, Cartesian, Cylindrical, Spherical

1. COORDINATE SYSTEM
Manju T. Kurian
AP, ECE
BM II CE

2. COORDINATE SYSTEM & TRANSFORMATION
• Coordinate systems are defined as a system used to represent a point
in space
• Classified a Orthogonal and Nonorthogonal Coordinate system
• For orthogonal coordinate system, the coordinates are mutually
perpendicular. The orthogonal coordinate systems include
• Rectangular or Cartesian coordinate system
• Cylindrical or circular coordinate system
• Spherical coordinate system

3. Rectangular Coordinate System
• This system is formed by three mutually orthogonal straight lines. The
three straight lines are called x,y,z axis. The point of intersection of
these lines are called the origin
• We will use the unit vectors ෞ
𝑎𝑥, ෞ
𝑎𝑦, ෞ
𝑎𝑧 to indicate the direction of
components along x,y,z axis respectively
• Range of coordinates x,y,z are −∞ < 𝑥 < ∞
−∞ < 𝑦 < ∞
−∞ < 𝑧 < ∞
• Vector Ԧ
𝐴 in cartesian coordinate can be written as
• Ԧ
𝐴=(Ax ,
Ay ,
A𝑧) =Ax ෞ
𝑎𝑥+ Ay ෞ
𝑎𝑦 + A𝑧ෞ
𝑎𝑧

4. • The position vector ത
𝒓 , a vector directed from the origin 0 to point P,
P(x,y,x) is given by
𝑂𝑃 = 𝑂𝐴 + 𝑂𝐵 + 𝑂𝐶
ҧ
𝑟 = 𝑥ෞ
𝑎𝑥 + 𝑦ෞ
𝑎𝑦 + 𝑧ෞ
𝑎𝑧
• x,y,z are scalar projections of ҧ
𝑟 𝑜𝑛 x,y and z axis
• Magnitude of ത
𝒓 is given by
ҧ
𝑟 = 𝑥2 + 𝑦2 + 𝑧2
• Orientation of ҧ
𝑟 on the x,y,z axis is given by 𝛼, 𝛽, 𝛾 𝑎𝑟𝑒 called Direction
Angles
ҧ
𝑟
𝑟
=
𝑥
𝑟
ෞ
𝑎𝑥 +
𝑦
𝑟
ෞ
𝑎𝑦 +
𝑧
𝑟
ෞ
𝑎𝑧
=Cos 𝛼 ෞ
𝑎𝑥+ Cos𝛽ෞ
𝑎𝑦+ Cos𝛾ෞ
𝑎𝑧
=𝑙𝑥 ෞ
𝑎𝑥+𝑙𝑦 ෞ
𝑎𝑦+ 𝑙𝑧ෞ
𝑎𝑧
𝑙𝑥, 𝑙𝑦, 𝑙𝑧 are known as Direction Cosines

5. • The unit vector along the direction of the position vector can be expressed in
terms of the direction cosines.
• Direction cosines are nothing but the magnitude of the position vector along the
coordinate axis
• Also Cos 𝛼 + Cos𝛽 + Cos𝛾 = 1
• Components in a specific direction
Consider a unit vector in a specific
direction.
The projection of Ԧ
𝐴 along the desired
direction is A.ො
𝑛 =A.Cos𝜃
This gives the magnitude of projection.
Scalar Component of Ԧ
𝐴 on 𝐵 = Ԧ
𝐴. ෞ
𝑎𝐵
Vector Component of Ԧ
𝐴 on 𝐵 =( Ԧ
𝐴. ෞ
𝑎𝐵) ෞ
𝑎𝐵

6. Relative Position Vector (Rij)
• Vector expression for Coulombs law, electric field intensity etc contains the
position vectors relative to the points other than the origin.
• Using the law of addition of vectors,
𝑟1 + 𝑟12 = 𝑟2
𝑟12 = 𝑅 = 𝑟2 − 𝑟1
• This can be considered as the position vector of (2) w.r.t position vector(1),
hence the name relative position vector
• The Unit vector of a relative position vector is
𝑎12=
𝑟12
𝑟12
=
𝑟2−𝑟1
𝑟2−𝑟1
, 𝑟2 − 𝑟1 = 𝑟1 − 𝑟2
Similarly, 𝑎𝑖𝑗 =
𝑟𝑗−𝑟𝑖
𝑟𝑗−𝑟𝑖

7. Cylindrical Coordinate System
• A point P in the cylindrical coordinate system is represented as
P(𝜌, ∅, 𝑧)
• Range of Coordinates
• Unit vectors
0≤ 𝜌 ≤ ∞
0≤ ∅ ≤ 2𝜋
-∞ ≤ 𝑧 ≤ ∞
𝑎𝜌 → 𝑎𝑙𝑜𝑛𝑔 𝜌 →Radius of the cylinder
𝑎∅ → 𝑎𝑙𝑜𝑛𝑔 ∅ →Azimuthal angle
𝑎𝑧 → 𝑎𝑙𝑜𝑛𝑔 𝑧 → 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑖𝑛 𝑐𝑎𝑟𝑡𝑒𝑠𝑖𝑎𝑛 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚

8. Spherical Coordinate System
• A point P in the Spherical coordinate system is represented as P(r, 𝜃, ∅)
• Range of Coordinates
• Unit vectors
0≤ 𝑟 ≤ ∞
0≤ 𝜃 ≤ 𝜋
0≤ ∅ ≤ 2𝜋
𝑎𝑟 → 𝑎𝑙𝑜𝑛𝑔 r →Radius of the sphere(meters)
𝑎𝜃 → 𝑎𝑙𝑜𝑛𝑔 𝜃 →Angle of elevation measured from z-axis (radians)
𝑎∅ → 𝑎𝑙𝑜𝑛𝑔 ∅ → Azimuthal Angle(𝑟𝑎𝑑𝑖𝑎𝑛𝑠)

9. Rectangular to Cylindrical Coordinate system
• 𝜌 = 𝑥2 + 𝑦2
• ∅ = 𝑡𝑎𝑛−1
(𝑦 ⁄ 𝑥)
• 𝑧 = 𝑧
• Sin ∅=
𝑦
𝜌
, Cos ∅=
𝑥
𝜌
, z=z
• X= 𝜌. Cos ∅
• y= 𝜌. Sin ∅
• Z=z
Cylindrical to Rectangular Coordinate system

10. Relationship between various coordinate
system
• (x,y,z)↔ 𝜌, ∅, 𝑧
• (x,y,z)↔ (𝑟, 𝜃, ∅)
𝜌 = 𝑥2 + 𝑦2 X= 𝜌. Cos ∅
∅ = 𝑡𝑎𝑛−1(𝑦 ⁄ 𝑥) y= 𝜌. Sin ∅
z=z z=z
𝑆𝑖𝑛𝜃 =
𝐶𝑃
𝑟
CP=r.Sin𝜃
Cos𝜃 =
𝑂𝐶
𝑟
=
𝑧
𝑟
z=r.Cos𝜃 = 𝑧
𝑆𝑖𝑛∅ =
𝐸𝐷
𝐶𝑃
=
𝑦
r.Sin𝜃
y= r.Sin𝜃. 𝑆𝑖𝑛∅
Cos∅ =
𝑂𝐸
𝐶𝑃
=
𝑥
r.Sin𝜃
x= r.Sin𝜃. Cos ∅
x= r.Sin𝜃. Cos ∅ r= 𝑥2 + 𝑦2 + 𝑧2
y= r.Sin𝜃. 𝑆𝑖𝑛∅ 𝜃=𝑡𝑎𝑛−1 𝑥2+𝑦2
𝑧
Z=r.Cos 𝜃 ∅=𝑡𝑎𝑛−1 𝑦
𝑥

11. • (𝜌, ∅, 𝑧)↔ 𝑟, 𝜃, ∅
𝜌=r.Sin 𝜃 r2= 𝜌2+z2
∅ = ∅ tan 𝜃 =
𝜌
𝑧
Z=r.Cos 𝜃 ∅ = ∅

12. Transformation Matrix
• Rectangular to Cylindrical Coordinate System
𝑎𝑥
𝑎𝑦
𝑎𝑧
=
𝐶𝑜𝑠∅ −𝑆𝑖𝑛∅ 0
𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0
0 0 1
𝑎𝜌
𝑎∅
𝑎𝑧
• Cylindrical to Rectangular Coordinate System
𝑎𝜌
𝑎∅
𝑎𝑧
=
𝐶𝑜𝑠∅ 𝑆𝑖𝑛∅ 0
−𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0
0 0 1
𝑎𝑥
𝑎𝑦
𝑎𝑧

13. Transformation Matrix
• Rectangular to Spherical Coordinate System
𝑎𝑥
𝑎𝑦
𝑎𝑧
=
𝑆𝑖𝑛𝜃. 𝐶𝑜𝑠∅ 𝐶𝑜𝑠𝜃. 𝐶𝑜𝑠∅ −𝑆𝑖𝑛∅
𝑆𝑖𝑛𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅
𝐶𝑜𝑠𝜃 −𝑆𝑖𝑛𝜃 0
𝑎𝑟
𝑎𝜃
𝑎∅
• Spherical to Rectangular Coordinate System
𝑎𝑟
𝑎𝜃
𝑎∅
=
𝑆𝑖𝑛𝜃. 𝐶𝑜𝑠∅ 𝑆𝑖𝑛𝜃. 𝑆𝑖𝑛∅ 𝐶𝑜𝑠𝜃
𝐶𝑜𝑠𝜃. 𝐶𝑜𝑠∅ 𝐶𝑜𝑠𝜃. 𝑆𝑖𝑛∅ −𝑆𝑖𝑛𝜃
−𝑆𝑖𝑛∅ 𝐶𝑜𝑠∅ 0
𝑎𝑥
𝑎𝑦
𝑎𝑧

14. DIFFERENTIAL VECTOR
• A differential vector or line element dl in Rectangular coordinate
system is given by
• ഥ
𝑑𝑙 = 𝑑𝑥ෞ
𝑎𝑥+ dyෞ
𝑎𝑦+ dzෞ
𝑎𝑧
𝑑𝑙 = 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2
• A differential volume element is given by dv=dxdydz

15. Differential vector in Cylindrical coordinate system
• A differential vector or line element dl in Cylindrical coordinate
system is given by
ഥ
𝑑𝑙 = 𝑑𝑙𝜌ෞ
𝑎𝜌+ 𝑑𝑙∅ෞ
𝑎∅+ 𝑑𝑙𝑧ෞ
𝑎𝑧
= 𝑑ρෞ
𝑎𝜌+ ρ𝑑∅ෞ
𝑎∅+ 𝑑𝑧ෞ
𝑎𝑧
• Differential arc length
𝑑𝑙 = (𝑑ρ)2+ (ρ𝑑∅)2+ (𝑑𝑧)2
• Differential area in the lateral surface of a cylinder
ρ𝑑∅𝑑𝑧. ෞ
𝑎𝜌
Differential in Coordinate Corresponding change in length
𝑑𝜌
𝑑∅
𝑑𝑧
𝑑𝑙𝜌=𝑑𝜌
𝑑𝑙∅=𝜌𝑑∅
𝑑𝑙𝑧=dz

16. Differential vector in Spherical coordinate system
• A differential vector or line element dl in Spherical coordinate system is
given by
ഥ
𝑑𝑙 = 𝑑𝑙𝑟ෞ
𝑎𝑟+ 𝑑𝑙𝜃 ෞ
𝑎𝜃 + 𝑑𝑙∅ෞ
𝑎∅
= 𝑑𝑟ෞ
𝑎𝑟+ 𝑟𝑑𝜃ෞ
𝑎𝜃+ 𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅ෞ
𝑎∅
• Differential arc length
𝑑𝑙 = (𝑑𝑟)2+ (𝑟𝑑𝜃)2+ (𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅)2
• Differential area on the Spherical Surface
𝑟𝑑𝜃𝑟.𝑆𝑖𝑛𝜃𝑑∅
• Differential Volume
(𝑑𝑟)(𝑟𝑑𝜃)(𝑟.𝑆𝑖𝑛𝜃𝑑∅)
Differential in Coordinate Corresponding change in length
𝑑𝑟
𝑑𝜃
𝑑∅
𝑑𝑙𝑟=𝑑𝑟
𝑑𝑙𝜃 =r𝑑𝜃
𝑑𝑙∅=𝑟. 𝑆𝑖𝑛𝜃. 𝑑∅

18. Curvilinear, Cartesian, Cylindrical, Spherical
Curvilinear Cartesian Cylindrical Spherical
Coordinate
𝑈1 x 𝜌 r
𝑈2 y ∅ 𝜃
𝑈3 z z ∅
Scale
factor
ℎ1 1 1 1
ℎ2 1 𝜌 r
ℎ3 1 1 r. Sin𝜃
Unit
Vector
𝑒1 ෞ
𝑎𝑥 ෞ
𝑎𝜌 ෞ
𝑎𝑟
𝑒2 ෞ
𝑎𝑦 ෞ
𝑎∅ ෞ
𝑎𝜃
𝑒3 ෞ
𝑎𝑧 ෞ
𝑎𝑧 ෞ
𝑎∅
Differential
length
𝑑𝑙1=ℎ1𝑈1 𝑑1 𝑑𝜌 𝑑𝑟
𝑑𝑙2=ℎ2𝑈2 𝑑2 𝜌 𝑑∅ r𝑑𝜃
𝑑𝑙3=ℎ3𝑈3 𝑑3 𝑑𝑧 r. Sin𝜃 𝑑∅