Coordinate Systems
https://emtmadeeasy.blogspot.com/unctions of space and time. In order to describe the spatial variations of the quantities, its important to define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. Hence its very important to understand the coordinate system first.
Coordinate system is a system of representing points in a space of given dimensions by coordinates, such as the Cartesian coordinate system or the system of celestial longitude and latitude.
Following are the links which will throw more light on the sub-topics related to co-ordinate systems in 3D and transformations:
- 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.
SOLVED EXAMPLES ON COORDINATE SYSTEM:
Q.1 Express the following points in Cartesian co-ordinate system.
a) P1 (2, 30o, 5)
b) P2 (4, 30o, 60o) SOLUTION/ANSWER
Q.2. Express the point P (1, -4, -3) in cylindrical and spherical co-ordinates? SOLUTION/ANSWER
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. SOLUTION/ANSWER
Q.4 Transform the vector E = (y2 – x2) ax + xyz ay + (x2 – z2) ax to cylindrical and spherical system? SOLUTION/ANSWER
Q.5 Express the vector A = ρ (z2 + 1)aρ - ρz cosφaφ in Cartesian co-ordinate system? SOLUTION/ANSWER
Q.6 Express the vector E =2r sinθ cosφar + r cosθ cosφaθ – r sinφaφ in Cartesian co-ordinate system?SOLUTION/ANSWER
Q.7 Calculate the distance between the following pair of points?
a) (2, 1, 5) and (6, -1, 2)
b) (3, π/2, -1) and (5, 3π/2, 5)
c) (10, π/4, 3π/4) and (5, π/6, 7π/4) SOLUTION/ANSWER
Q.8 Find the distance between A (2, π /6, 0) and B = (1, π /2, 2) ? SOLUTION/ANSWER
Q.9 Using the differential length dl, for the length of each of the following curves:
a) ρ = 3, π /4 < φ < π /2, z = constant.
b) r = 1, θ = 30o, 0o < φ < 60o
c) r = 4, φ = constant, 30o < θ < 90o SOLUTION/ANSWER
Q.10 Calculate the areas of the following surfaces using the differential surface area ds:
a) ρ = 2, π/3 < φ < π/2, 0 < z < 5.
b) r = 10, π/4 < θ < 2π/3 , 0 < φ < 2π SOLUTION/ANSWER
Q.11 Use the differential volume dv to determine the volumes of the following regions:
a) 0 < x < 1, 1 < y <2, -3 < z < 3
b) 2 < ρ < 5, π/3 < φ < π, - 1 < z < 4 SOLUTION/ANSWER
Coordinate system is a system of representing points in a space of given dimensions by coordinates, such as the Cartesian coordinate system or the system of celestial longitude and latitude.
Following are the links which will throw more light on the sub-topics related to co-ordinate systems in 3D and transformations:
- 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.
SOLVED EXAMPLES ON COORDINATE SYSTEM:
Q.1 Express the following points in Cartesian co-ordinate system.
a) P1 (2, 30o, 5)
b) P2 (4, 30o, 60o) SOLUTION/ANSWER
Q.2. Express the point P (1, -4, -3) in cylindrical and spherical co-ordinates? SOLUTION/ANSWER
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. SOLUTION/ANSWER
Q.4 Transform the vector E = (y2 – x2) ax + xyz ay + (x2 – z2) ax to cylindrical and spherical system? SOLUTION/ANSWER
Q.5 Express the vector A = ρ (z2 + 1)aρ - ρz cosφaφ in Cartesian co-ordinate system? SOLUTION/ANSWER
Q.6 Express the vector E =2r sinθ cosφar + r cosθ cosφaθ – r sinφaφ in Cartesian co-ordinate system?SOLUTION/ANSWER
Q.7 Calculate the distance between the following pair of points?
a) (2, 1, 5) and (6, -1, 2)
b) (3, π/2, -1) and (5, 3π/2, 5)
c) (10, π/4, 3π/4) and (5, π/6, 7π/4) SOLUTION/ANSWER
Q.8 Find the distance between A (2, π /6, 0) and B = (1, π /2, 2) ? SOLUTION/ANSWER
Q.9 Using the differential length dl, for the length of each of the following curves:
a) ρ = 3, π /4 < φ < π /2, z = constant.
b) r = 1, θ = 30o, 0o < φ < 60o
c) r = 4, φ = constant, 30o < θ < 90o SOLUTION/ANSWER
Q.10 Calculate the areas of the following surfaces using the differential surface area ds:
a) ρ = 2, π/3 < φ < π/2, 0 < z < 5.
b) r = 10, π/4 < θ < 2π/3 , 0 < φ < 2π SOLUTION/ANSWER
Q.11 Use the differential volume dv to determine the volumes of the following regions:
a) 0 < x < 1, 1 < y <2, -3 < z < 3
b) 2 < ρ < 5, π/3 < φ < π, - 1 < z < 4 SOLUTION/ANSWER
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