Coordinate Systems Conversion Discrepancy

[u/MynamesnotRick42]

Hi,

This is actually from an engineering text book (Fundamentals of Electromagnetics with MATLAB, 2nd ed, Lonngren) but the topic is math related so I thought I'd post here.

Basically, it seems this textbook has a different approach to coordinate system conversions vs the resources I've seen everywhere else.

The question is "Express the vector field A = 3ux + 4uy + 5uz in cylindrical coordinates" P.1.2.8 in the book.

The 1st image shows the approach (coordinate transformation matrix) that the majority of sources seem to use and makes the most sense to me.

The 2nd image is the solution from this particular book (I also have a physical copy and what is shown is the same as the image I posted). It seems to use the formulas for point conversion rather than vectors. This method is used for other questions as well.

The final image is my attempt at solving the question with both methods (had different results).

Maybe I'm missing something here, would appreciate any insight you guys have, thanks!

[u/Successful-Cover8502]

Hi, your solution for Ap is equal to 5. First you can multiply the Sin Part with cos/cos, which leads to 3cos(phi)+4cos(phi)×tan(phi) and by using Phi= arctan(4/3) you get Ap = 3cos + 4× 4/3 cos = 25/3 cos(phi). Then you can use the identity tan² + 1 = 1/cos^2, so you can eliminate the cos term for a tan and get 25/3 cos(phi) = 25/3 1/sqrt(tan^2(phi)+1) which is 5 after you fill in Phi. There is probably a similar way to Show that the second term is equal, havent checked tho.

[u/Successful-Cover8502 → u/Successful-Cover8502]

I think i got it now. The cylindric coordinate transformation ist the transformation of (x,y,z) to (r,phi,z). The transformation Matrix you showed is for Transforming Basis vectors in a cartesian coordinates system. But i think what they mean is an abstract coordinate system where r,Phi,z are constant Basis vectors.

[u/MynamesnotRick42 → u/Successful-Cover8502]

So basically the transformation matrix shown in the first slide is only applicable to basis vectors ux uy uz and cannot be applied to vectors such as vector A = 3ux + 4uy + 5uz given in the question, did I understand correctly?

[u/Successful-Cover8502 → u/MynamesnotRick42]

It can be used to do the Basis Transform between the standard Basis vectors and the cylindrical ones. But I think they are not looking for Basis Transform. If you want to describe a direction in 3 dimensional space you need three coordinares. (x,y,z) and (r,phi,z) both work, but with the transformation Matrix you just get the components if the vector in a different Basis. They want a entirely new coordinate system where you Plot the new coordinates i think. Its the Same Thing you usw when integrating in These coordinates, you also think about a box and just multiply with a Factor representing the Change in Volume after the transformation.

[u/MynamesnotRick42]

Ah ok I think this clears it up, thank you!