Griffiths example no. 5.11 w×r switch from cartesian to spherical

[sudipmaity]

I am pretty much satisfied with the example of a rotating shell example 5.11 pg 367 griffiths electrodynamics.on many ocassions he chooses cartesian coordinates before integration (see 5.10 too) , integrates and finds w×r along y direction .then he manipulates w×r, and writes it down in spherical polar along phi cap or the azimuth.
MY QUESTION IS IT REALLY NECESSARY TO MAKE THIS SWITCH? HOW DOES IT SERVE THE PURPOSE OF FURTHER CALCULATION? A lot of books which solved this problem avoided this switch.BESIDES HOW IS HE DOING THIS SWITCH? CAN ANYBODY GIVE ME ANY REFERENCE OF AN EXAMPLE WHERE IT IS SHOWN THAT -ωsinψy in cartesian when changed to spherical system becomes ωrsinθ∅

 

[Nate810]

?It is not necessary to make the switch from Cartesian coordinates to spherical polar coordinates in order to solve the rotating shell example 5.11 in Griffiths Electrodynamics. The switch is often made in order to simplify the calculations and make the solution easier to visualize. The switch from Cartesian coordinates to spherical polar coordinates can be done using the standard transformations, which are given by: x = rsinθcosΦy = rsinθsinΦz = rcosθWhere x, y, and z are the Cartesian coordinates and r, θ, and Φ are the spherical polar coordinates. For example, if one has a vector in Cartesian coordinates of the form ω = (0, ωsinψ, 0)the same vector can be written in spherical polar coordinates asω = (0, ωrsinθcosΦ, 0)By substituting the standard transformations into the expression for the vector, it is possible to derive the relationship between the Cartesian and spherical polar components of the vector. Hope this helps!