# Cylindrical coordinates: unit vectors and time derivatives

## Homework Statement ## 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?

## Answers and Replies

PeroK
Science Advisor
Homework Helper
Gold Member
2020 Award
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
Mentor

## Homework Statement

View attachment 237940

## 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! 