Velocity transformation from spherical to cartesian coords

[Stollaxel Stoll]

I cant't figure out how to transform $\dot{r}$, $\dot{\theta}$, $\dot{\phi}$ in spherical coordinates to $\dot{x}$, $\dot{y}$, $\dot{z}$ in cartesian coordinates (the dot is Newton's notation for the first time-derivative which is the angular velocity and velocity).

I have no trouble transforming the coordinates, but if I try $\sqrt{\dot{r}^2+(\dot{\theta} r)^2+(\dot{\phi} r)^2}$ I get the wrong total velocity if adding up the components by Pythagoras. Any ideas why this doesn't work, and even more important, how it works instead?

 

[mathman]

https://en.wikipedia.org/wiki/Spherical_coordinate_system

See above to get the correct form.

 

[Yukterez]

but if I try $\sqrt{\dot{r}^2+(\dot{\theta} r)^2+(\dot{\phi} r)^2}$ I get the wrong total velocity if adding up the components by Pythagoras. Any ideas why this doesn't work, and even more important, how it works instead?

The Wikipedia article is overloaded with unnecessary complicated notation. You just forgot to that the longitude diameter is 2πr, but the latitude diameter depends on the longitude so change your Pythagoras to $|v| = \sqrt{\dot{r}^2+(\dot{\theta} r)^2+(sin(\theta) \dot{\phi} r)^2}$