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

  • upload_2019-1-28_14-39-18.png
    upload_2019-1-28_14-39-18.png
    18.1 KB · Views: 519
  • upload_2019-1-28_14-40-20.png
    upload_2019-1-28_14-40-20.png
    13.3 KB · Views: 262
  • upload_2019-1-28_14-44-58.png
    upload_2019-1-28_14-44-58.png
    4.1 KB · Views: 178
  • upload_2019-1-28_14-46-30.png
    upload_2019-1-28_14-46-30.png
    4.1 KB · Views: 355

Answers and Replies

  • #2
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2020 Award
17,125
8,942
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
21,158
4,672

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?
 
  • Like
Likes Mason Smith
  • #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 on Cylindrical coordinates: unit vectors and time derivatives

Replies
9
Views
922
Replies
6
Views
3K
Replies
1
Views
4K
Replies
2
Views
560
  • Last Post
2
Replies
37
Views
7K
Replies
9
Views
894
Replies
6
Views
42K
Replies
2
Views
21K
Top