• Support PF! Buy your school textbooks, materials and every day products Here!

Cylindrical coordinates: unit vectors and time derivatives

  • #1

Homework Statement


upload_2019-1-28_14-39-18.png


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.
upload_2019-1-28_14-40-20.png

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.
upload_2019-1-28_14-46-30.png

Can someone enlighten me, please?
 

Attachments

Answers and Replies

  • #2
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
11,744
4,721
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.
 
  • #3
19,799
4,048

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?
 
  • #4
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! :smile:
 

Related Threads for: Cylindrical coordinates: unit vectors and time derivatives

Replies
1
Views
4K
Replies
0
Views
2K
Replies
3
Views
8K
Replies
9
Views
816
Replies
9
Views
714
Replies
6
Views
2K
Replies
6
Views
40K
Replies
4
Views
1K
Top