How do you change Curl in Mathematica so that it uses theta as latitude rather than as colatittude? Changing theta to Pi/2 - theta doesn't seem to get all the signs right in the final result.
Basically, I want to use the curl in a right handed spherical coordinate system with unit vectors going eastward, northward, and vertical. If I take the Curl[R[r, theta, phi]], where R is a vector{R, Theta, Phi} and change theta to Pi/2 - theta so that the angle used in the result from Mathematica's Curl is measured from the equator instead of the pole (i.e., it is now the Latitude instead of coLatitude), the result does not quite agree with what I find documented in the literature. Two of the three vector components agree in terms and signs but the third Mathematica component gives
Tan[theta]*Theta[r, theta, phi]/r Sec[theta]*D[Theta[r, theta, phi], phi]/r +
Phi[r, theta, phi]/r
whereas the literature gives
Tan[theta]*Theta[r, theta, phi]/r + Sec[theta]*D[Theta[r, theta, phi], phi]/r -
Phi[r, theta, phi]/r
which agrees in terms but not in sign. Multiplying by -1 still leaves one term in disagreement.
So, I guess a better form of the question is how do you change the Mathematica curl from its current Eastward, Southward, Vertical coordinate system to an Eastward, Northward, Vertical coordinate system that uses Latitude rather than colatitude for theta.?
