I think I authored a similar Thread here or at an other forum, but never got an answer I trusted.

There are well known equations for transforming a Cartesian point (xyz) to Spherical coordinates and vice versa. What are the equations for transforming velocity and/or acceleration vectors?

A while ago, I assumed that taking derivatives of the point conversion formulae would provide correct equations. Then an engineer claimed that a Cartesian point (x, y, z) is a position vector. He further stated that a vector is a vector is a vector. Hence, the equations for transforming a position vector should be correct for transforming velocity and acceleration vectors.

I considered the possibility that both methods were correct and would arrive at the same results. Some work with my MathCad7 software convinced me that this was not true. The two methods do not lead to the same results.

Does anybody have some thoughts on this? Ignoring the formulae for Longitude and Latitude, consider the following equation for the Spherical coordinate radius.

r = SquareRoot( x2 + y2 + z2 )

Derivative analysis results in the following for transforming velocity.

Vr = ( x*Vx + y*Vy + z*Vz ) / r, where Vr, Vx, Vy, Vz are velocities.

Using the position vector formula on the velocity vector results in the following.

Vr = SquareRoot(Vx2 + Vy2 + Vz2 )

Which of the above is correct for Vr? If you have an opinion, please explain.

My vote goes for the first equation, but I am not sure.