Solved Examples/Numericals On Coordinate System & Transformation - 1.

Q.1 Express the following points in Cartesian co-ordinate system.
a) P1 (2, 30^o, 5) 
b) P2 (4, 30^o, 60^o)

Ans.
a) P₁ (ρ, φ, z) → P₁ (x, y, z)
x = ρ cosφ = 1cos 60^o = 0.5
y = ρ sinφ = 1sin 60^o = 0.87
z = z = 2

Therefore P₁ = (0.5, 0.87, 2)

b) P₂ (r, θ, φ) → P₂ (x, y, z)
x = r sinθ cosφ = 4sin 30^o cos 60^o = 1
y = r cosθ cosφ = 4cos 30^o cos 60^o= 1.73
z = r cosθ = 4cos 60^o = 3.46

Therefore P₂ = (1, 1.73, 3.46)



Q.2. Express the point P (1, -4, -3) in cylindrical and spherical co-ordinates?

Ans.
P₁ (x, y, z) → P₁ (ρ, φ, z)
P₂ (x, y, z) → P₂ (r, θ,φ)

Q. 3 
a) If V = XZ – XY +YZ, express V in Cylindrical co-ordinate system.

b) If U = X² + 2Y² +3z², express U in Spherical co- ordinates System.

Ans.
a) Since the equation given is a scalar equation, hence we just need to substitute the values of x, y and z in terms of ρ, φ and z.

We know x = ρ cosφ and y = ρ sinφ

V= xz – xy – yz
= (ρ cosφ) z – (ρ cosφ) (ρ sinφ) – (ρ sinφ)z
= ρz cosφ – ρ² cosφ sinφ – ρz sinφ


b) We know x = r sinθcosφ; y = r sinθsinφ & z = rcosθ

U = x² + 2y² +3z²
= (r sinθcosφ)² +2(r sinθsinφ)² +3(r cosθ)²
= r² + r² sin² θsin² φ + 2 r²cos²θ



Q.4 Transform the vector E = (y² – x²) a_x + xyz a_y + (x² – z²) a_x to cylindrical and spherical system?

Ans.

E_ρ = (y² – x²) cosφ + xyz sinφ
= ρ² (sin² φ - cos² φ) + ρ² z cosφ sin² φ
= - ρ² cos2φ cosφ + ρ² z cosφ sin² φ

E _φ = - (y² – x²) sinφ + xyz cosφ
= ρ² cos2φ cosφ + ρ² z cos² φ sin φ

E_z = x² – z² = ρ² cos² φ - z²

E = ρ² cosφ (z sin² φ - cos2φ)a_ρ + ρ² sinφ (z cos² φ + cos2φ)a_φ
+ (ρ² cos² φ - z²)a_z

Now E in spherical co-ordinate system is given as ,

E_r = (y² – x²) sinθ cosφ + xyz sinθ sinφ + (x² – z²) cosθ

= (r² sin² θsin² φ – r² sin² θcos² φ) sinθ cosφ +(r sinθcosφ)(r sinθsinφ)(r cosθ) sinθ sinφ + ( r² sin² θcos² φ – r² cos² θ) cosθ

= r² sin³ θ (sin² φ - cos² φ) cosφ + r³ sin³ θsin² φ cosθcosφ + ( r² sin² θcos² φ – r² cos² θ) cosθ

Similarly,

E_θ = - r² sin² θ (cos2φ) cosθ cosφ + r² sin² θsin² φ cos² θcosφ - (r² sin² θcos² φ – r² cos² θ) sinθ

E_φ = -(r² sin² θsin² φ – r² sin² θcos² φ) sinφ + r³ sin² θsinφ cosθ cos² φ

E = E_r a_r + E_θ a_θ + E_φ a_φ 




Q.5 Express the vector A = ρ (z² + 1)a_ρ - ρz cosφa_φ in Cartesian co-ordinate system?
Ans.

A_x = ρ (z² + 1) cosφ + ρz cosφ sinφ

A_y = ρ (z² + 1) sinφ - ρz cos²φ

A_z = 0
Hence, A = A_xa_x + A_ya_y + A_zaz

Q.6 Express the vector E =2r sinθ cosφa_r + r cosθ cosφa_θ – r sinφa_φ in Cartesian co-ordinate system?
Ans.

E_x = 2r sin² θcos² φ + r cos² θcos² φ + r sin² φ

Substituting the above values, we get

ALSO READ:

- Introduction To Coordinate System.

- Cartesian Coordinate System / Rectangular Coordinate System (x, y, z).

- Differential Analysis Of Cartesian Coordinate System.

- Circular Cylindrical Coordinate System (ρ, φ, z).

- Differential Analysis Of Cylindrical Coordinate System.


- Spherical Coordinate System ( r, θ , φ).

- Differential Analysis Of Spherical Coordinate System.

- Numericals / Solved Examples - Page 1.

- Numericals / Solved Examples - Page 2.


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