How to transform rectangular coordinate velocity into polar coordinate velocity In a three-dimensional velocity field, it is necessary to decompose the velocity on a circular surface into radial velocity and tangential velocity We know how to use the dot product of vector to get the radial velocity, but we don't know how to get the tangential velocity. Because velocity has direction, we can't simply use Pythagorean theorem to get the tangential velocity Can you explain in detail how to find the tangent vector of the next derivative

How to transform rectangular coordinate velocity into polar coordinate velocity In a three-dimensional velocity field, it is necessary to decompose the velocity on a circular surface into radial velocity and tangential velocity We know how to use the dot product of vector to get the radial velocity, but we don't know how to get the tangential velocity. Because velocity has direction, we can't simply use Pythagorean theorem to get the tangential velocity Can you explain in detail how to find the tangent vector of the next derivative

The radial velocity at a point is equal to the inner product of the unit radial vector and the velocity at that point. The same principle: the tangential velocity at a point is equal to the inner product of the unit tangential vector and the velocity at that point. As for the tangential vector, because it is a circular motion, we can use the natural parameter (i.e. arc length parameter) to set up the expression of the circle, or use polar coordinates