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Converted document EXAMPLES: SPHERICAL COORDINATES

EXAMPLES: SPHERICAL COORDINATES

Given a cartesian vector of the form
(1) Xi + Yj + Zk
and a spherical vector of the form
(2) rer + φeφ + θeθ
Conversion from spherical to Cartesian coordinates can be accomplished by
(3) X  = rsinφcosθ     Y  = rsinφsinθ     Z  = rcosφ
Conversion back to spherical coordinates can be perfomed using
(4) r  = (X2 + Y2 + Z2)(1 ⁄ 2)     φ  = cos − 1⎛⎝(Z)/(r)⎞⎠     θ  = ⎧⎨⎩ sin − 1⎛⎝(Y)/((X2 + Y2)(1 ⁄ 2))⎞⎠   if X ≥ 0        π − sin − 1⎛⎝(Y)/((X2 + Y2)(1 ⁄ 2))⎞⎠   if X < 0 
Example 1
Find the spherical coordinates of the point whose Cartesian coordinates are (4, 6, 5).
To find the spherical coordinates, plug the Cartesian coordinates into into equation 4↑.
(5) r  = (42 + 62 + 52)(1 ⁄ 2) ≈ 8.7750     φ  = cos − 1⎛⎝(5)/(8.7750)⎞⎠ ≈ 0.9645rad     θ  = sin − 1⎛⎝(6)/((42 + 62)(1 ⁄ 2))⎞⎠ ≈ 0.9828rad
The spherical representation of the Cartesian vector 4 i + 6 j + 5 k is therefore
(6) 8.7750 er + 0.9646 eφ + 0.9828 eθ.
Example 2
Find the Cartesian coordinates of the point whose spherical coordinates are (10, pi/6, pi/3).
To find the Cartesian coordinates, plug these spherical coordinates into into equation 3↑.
(7) X  = 10sin⎛⎝(π)/(6)⎞⎠cos⎛⎝(π)/(3)⎞⎠ = 2.5     Y  = 10sin⎛⎝(π)/(6)⎞⎠sin⎛⎝(π)/(3)⎞⎠ ≈ 4.3301     Z  = 10cos⎛⎝(π)/(6)⎞⎠ ≈ 8.6603