Cylindrical coordinates: unit vectors and time derivatives

[Mason Smith]

Homework Statement

Homework Equations

The Attempt at a Solution

I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates.

I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am having trouble seeing how it simplifies. For instance, I do not understand how to arrive at the following for the rho hat unit vector.

Can someone enlighten me, please?

 

[PeroK]

You say you have the time derivatives of the unit vectors. But, if the derivative of $\hat{\rho}$ is not that given, then you must have made a mistake.

 

[Chestermiller]

Homework Statement

View attachment 237940

Homework Equations

The Attempt at a Solution

I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates.
View attachment 237941
I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am having trouble seeing how it simplifies. For instance, I do not understand how to arrive at the following for the rho hat unit vector.
View attachment 237943
Can someone enlighten me, please?

You have $$\dot{\hat{\rho}}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})\dot{\phi}$$But you already showed that $$\hat{\phi}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})$$
Do you see how it works out now?

 

[Mason Smith]

You have $$\dot{\hat{\rho}}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})\dot{\phi}$$But you already showed that $$\hat{\phi}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})$$
Do you see how it works out now?

That makes perfect sense. Thank you so much for the insight, Chestermiller!