Answer :
Equation representing the conversion of spherical coordinates to rectangular coordinates are r = √ x² + y² + z² , φ = cos⁻¹( z/r) ,
θ = tan⁻¹(y/x) and Jacobian transformation is given by :
[tex]d(x, y, z)/ d(r, \theta, \phi ) =\left|\begin{array}{ccc}dx/dr&dx/d\theta&dx/d\phi\\dy/dr&dy/d\theta&dy/d\phi\\dz/dr&dz/d\theta&dz/d\phi\end{array}\right|[/tex]
As given in the question,
In the spherical coordinates are ( r, θ, φ ) to the given rectangular coordinates in the cartesian plane are ( x, y , z)
Equation representing the conversion of spherical and rectangular coordinates is given by:
x = rcosθsinφ
y = rsinθsinφ
z = rcosφ
Where
0 ≤ r <∞ , 0≤ θ< 2π , 0≤φ< π
r = √ x² + y² + z²
φ = cos⁻¹( z/r)
θ = tan⁻¹(y/x)
Jacobian transformation is given by:
[tex]d(x, y, z)/ d(r, \theta, \phi ) =\left|\begin{array}{ccc}dx/dr&dx/d\theta&dx/d\phi\\dy/dr&dy/d\theta&dy/d\phi\\dz/dr&dz/d\theta&dz/d\phi\end{array}\right|[/tex]
Where,
dx/dr = cosθsinφ
dx/dθ = -rsinθsinφ
dx/dφ = rcosθcosφ
dy/dr = sinθsinφ
dy/dθ = rcosθsinφ
dy/dφ = rsinθcosφ
dz/dr = cosφ
dz/dθ = 0
dz/dφ = -rsinφ
Therefore, the conversion of spherical coordinates to rectangular coordinates are given by : r = √ x² + y² + z² , φ = cos⁻¹( z/r) ,θ = tan⁻¹(y/x) and Jacobian transformation is given by :
[tex]d(x, y, z)/ d(r, \theta, \phi ) =\left|\begin{array}{ccc}dx/dr&dx/d\theta&dx/d\phi\\dy/dr&dy/d\theta&dy/d\phi\\dz/dr&dz/d\theta&dz/d\phi\end{array}\right|[/tex]
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