Question

Three frequently used coordinates in R

3 are 1: Cartesian Coordinates, 2: Cylindrical coordi-

nates, and 3: Spherical Coordinates. Unit vectors in these coordinates are the following.

Cartesian coordinates (x, y, z):

eˆx = ˆi, (1)

eˆy = ˆj, (2)

eˆz = ˆk. (3)

Cylindrical coordinates (ρ, φ, z):

eˆρ = cos φ ˆi + sin φ ˆj, (4)

eˆφ = − sin φ ˆi + cos φ ˆj, (5)

eˆz = ˆk (6)

Spherical coordinates (r, θ, φ):

eˆr = sin θ cos φ ˆi + sin θ sin φ ˆj + cos θ

ˆk, (7)

eˆθ = cos θ cos φ ˆi + cos θ sin φ ˆj − sin θ

ˆk, (8)

eˆφ = − sin φ ˆi + cos φ ˆj. (9)

a) Find the expressions of (ˆer, eˆθ, eˆφ) in terms of (ˆeρ, eˆφ, eˆz).

b) A vector can be expressed in terms of any basis. In particular, we can write:

V~ = Vxeˆx + Vyeˆy + Vzeˆz = Vρeˆρ + Vφeˆφ + Vzeˆz = Vreˆr + Vθeˆθ + Vφeˆφ. (10)

In terms of column vector notation we write

VCr =





Vx

Vy

Vz



 , VCl =





Vρ

Vφ

Vz



 , VSp =





Vr

Vθ

Vφ



 . (11)

The relations between them can be expressed in terms of matrix:

VCr = PCr←ClVCl, VCr = PCr←SpVSp, (12)

VCl = PCl←CrVCr, VCl = PCl←SpVSp, (13)

VSp = PSp←CrVCr, VSp = PSp←ClVCl, (14)

where PCr←Cl, PCr←Sp, PCl←Cr, PCl←Sp, PSp←Cr, PSp←Cl are all 3×3 matrices. Without explicitly

finding the matrices show that:

PSp←CrPCr←Sp = I3, PSp←Cl = PSp←CrPCr←Cl. (15)

c) Now explicitly compute PSp←Cl and show that PSp←Cl ∈ SO(3)​

Answer 1

Answer:

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