If (x,y,z) is rectangular Cartesian coordinates, and ⁽theta¹,theta²,theta³⁾, is the new curvilinear coordinates defined by the transformation
x=theta¹theta²cotheta³
y=theta¹theta²intheta³
z=1/2 [⁽theta¹⁾²⁻⁽theta²⁾²]
Please determine:
(a)The unitary basis {¹⇀,ᵛᵉᶜ⁽⁾²,ᵛᵉᶜ⁽⁾³}
in terms of the Cartesian basis {ℎ(),hat(),hat()}(
(b) Is this transformation orthogonal or not?
(c)The scalar factors in this coordinates,
(d)The components of metric tensor gᵢⱼ,
(e)The physical components of a vector,
(f)The dvec() in terms of ⁽theta¹,theta²,theta³⁾,
(g)The arc length ds,
(h)The volume element in terms of ⁽theta¹,theta²,theta³⁾,
(i)The differential angle of rotation dvec()
(j)The partial derivatives of the unitary basis vector: Please only solve the assigned part above.