Tutorials on co-ordinate systems

1. Kathmandu Engineering College, Kalimati, Kathmandu
Subject: Electromagnetics
Tutorial Sheet: 1
1. a) Give rectangular co–ordinates of the point C(ρ = 4.4, ϕ = −115°, z = 2). Specify C in Cartesian Co-ordinate.
b) Give cylindrical co–ordinates of the point D(x = −3.1, y = 2.6, z = −3). Specify C in Cylindrical Co-ordinate.
c) Specify the distance from C to D.
Ans: a) (−1.850, − 3.9877,2)b) (4.045, 140°,-3) c) 8.364
2.a) F⃗ = 10âx − 8ây + 6âz at point P(10, − 8, 6). Ans: 12.82â + 6âz
b) G⃗ = (2x + y)âx − (y − 4x)ây at point Q(ρ, ϕ, z)
Ans:a) 12.82â + 6âz b) 2ρ cos2
ϕ + 5ρ sin ϕ cos ϕ − ρ sin2
ϕ â + 4ρ cos2
ϕ − 3ρ cos ϕ sin ϕ − ρ sin2
ϕ â
3.Express the vector that extends from P(−2, − 5, 1) to Q(5, − 4, 8) in cylindrical co–ordinate.
Ans:(7 cos ϕ + sin ϕ)âx + (−7 sin ϕ + cos ϕ)ây + 7âz
4.Express the vector field W⃗ = (x − y)ây in cylindrical and spherical co–ordinates.
Ans: r sin θ (cos ϕ − sin ϕ) sin θ sin ϕ âr + cos ϕ sin ϕ â + cos ϕ â
5.Transform vector A⃗ = ρ sin ϕ âz at point (1, 45°, 2) in cylindrical co–ordinate system to a vector in spherical co–
ordinate system. Ans:âr − 1.414â
6.Transform A⃗c = xâx + xyâz at point (1, 2, 3) in Cartesian co–ordinate system to A⃗cy in cylindrical co–ordinate
system.Ans:0.447â − 0.894â + 2âz
7.Transform the vector B⃗ = yâx − xây + zâz into cylindrical co–ordinates.Ans: −ρâ + zâz
8.Transform vector field A⃗ = ρ cos ϕ â + zâz at point P(1, 30°, 2) in cylindrical co–ordinate system to spherical co–
ordinate system.Ans: 2.175âr − 0.113â
9.Express the vector fieldD⃗ = in cylindrical components and cylindrical variables. Ans: 0.5â
10. Transform the following vectors
a) A⃗ = yâx + (x + y)âz at P(−2, 6, 3) into cylindrical system. Ans: 1.897âx + 5.69ây + 4âz
b) A⃗ = 3â − 4â atQ(3, 4, − 1) into Cartesian system. Ans: 5âx
11.Transform the vector B⃗ = ya + xa + za into cylindrical co–ordinate. Ans:−ρa∅ + za
12. Given a point P(−3, 4, 5), express the vector that extends from P to Q(2, 0, − 1) in (a) rectangular co–ordinates
(b) cylindrical co–ordinates (c) spherical co–ordinates.
Ans: a)5âx − 4ây − 6âz b)(5 cos ϕ − 4 sin ϕ)â − (5 sin ϕ + 4 cos ϕ)â − 6âz c)
(5 sin θ cos ϕ − 4 sin θ sin ϕ − 6 cos θ)âr + (5 cos θ cos ϕ−4 cos θ sin ϕ + 6 sin θ)â − (5 sin ϕ + 4 cos ϕ)â
13.At a point (−3, −4,5) transform the vector to spherical co-ordinate system that extends from P to
(2,0, −1)Ans: -8.63âr-0.141âθ+1.6â∅
14. Transform the point given in rectrangular co-ordinate system to cylindrical and spherical co-
ordinate system.
15. Derive the transformation matrix to transform:
a) Cartesian co-ordinate system to cylindrical co-ordinate system & vice-versa
b) Cartesian co-ordinate system to spherical co-ordinate system& vice-versa
c) Cylindrical co-ordinate system to spherical co-ordinate system & vice-versa
16. Express the differential length, differential area and differential volume for the rectangular,
cylindrical and spherical co-ordinate system.