Conversion from Cartesian to Cylindrical Coordinate System

Hello there,

I was solving problem 1.14 in Schaum's Outlines Electromagnetics (2nd Edition).

The problem states to convert a vector A from Cartesian to Cylindrical coordinates. I am attaching the problem and my attempt towards it. The last image shows how the book solved it. My solution is way off and I can't seem to grasp the one in the book. Please let me know what I am doing wrong and also what the book is doing in the section I circled red.

Question:



My Attempt:



Book Solution:



Reference image:




Thanks for the help

x, y and z are not constants. Imagine the problem as vectors on set of points, not a vector from (0,0,0).

Dear FootTea,

Thank you for your reply. I understand that because x, y, z are not constants we can not just use them directly. However, I am still having a hard time in understanding the "projection" concept. Is there some other way of doing it?

Thanks, I appreciate your help

They are just using dot product. Dot product of two vectors equals the product of the magnitude of the two vectors and the cosine of the angle between them.

Your basic answer is correct. Actually the book answer is confusing.

The book answer is confusing because it is not expressed in terms of x,y,z

What is the cylindrical coordinates for (x,y,z)? The correct answer is (sqrt(x*x+y*y), arctan(y/x),z)

The book answer is (r,phi,z)- which is silly.

c_mitra said:

The book answer is (r,phi,z)- which is silly.

Click to expand...

The answer given by the book is not silly.
The coordinates are expressed in terms of the cyindrical transformation versos (.e. unity vectors).
This mens those versors are associated to the cylindrical space. The transormation is from (ax, ay, az) to (ar, aφ, az).

A versor indicates the direction of the axis, then it's defined as the derivative along the wanted axis divided by its modulus.

In our case we want the direction along R, (here capitals letter stands for direction) with respect to r, φ and z.

We have R=[r*cos(φ), r*sin(φ), z] thus:

(dR/dr)/|dR/dr| = =[cos(φ), sin(φ), 0]
(dR/dφ)/|dR/dφ| = =[-sin(φ), cos(φ), 0]
(dR/dz)/|dR/dz| = =[0, 0, 1]

then:

ax = cos(φ)*ar - sin(φ)*aφ
ay = sin(φ)*ar + cos(φ)*aφ
az = az

albbg said:

The answer given by the book is not silly...

Click to expand...

I beg your pardon?

Please see the original question: the coordinates are given in terms of x and y (z does not matter because it does not get transformed)

(x,y) in cartesian coordinates is ((x*x+y*y)^0.5, arctan(y/x)) in polar coordinates.

It is like asking "what is the color of the sky?" and the answer is "cow has four legs"

I say "your answer is wrong" and you say "does n't cow has four legs?"

Sorry, but I didn't understand what you want to say.

The original question is speaking about a vector "A". Its orientation is given by the versors (ax, ay, az) in the same way, transforming we will have a vector "At" referred to the cyclindrical basis: its orientation is represented by (ar, aφ, az).

That is A=f(x,y,z)*ax+g(x,y,z)*ay+h(x,y,z)*az ==> transform to ==> At=ft(r,φ,z)*ar+gt(r,φ,z)*aφ+ht(r,φ,z)*az

Where is the mismatch ?

Albbg is correct. When you're converting from Cartesian to Cylindrical coordinate systems you should not have any x's or y's in your answer. Your answer should only be in terms of r, φ, and z. Remember this is converting between coordinate systems not just points.

@dzafar You have the right idea but you're basically going about it backwards. I'm a little rusty but what I remember is first you use your identities to remove all the x's and y's. The identities are the x = rcos(φ) ones. Once you've done that you should end with something with only r, φ, and z. But you will still have the Cartesian unit vectors which you need to replace with the cylindrical unit vectors. That's where all those dot products are coming in. When I did this I just used the matrix multiplication(You can find it on the Wiki) to do this. Or you can use those equations provided below the dot products. Substitute everything in. Use the dot product info to multiple unit vectors and you're done.

Hopefully that helps. I don't know Latex so my explanations are a little weird so as to avoid notation.

albbg said:

Sorry, but I didn't understand what you want to say.

The original question is speaking about a vector "A". Its orientation is given by the versors (ax, ay, az) in the same way, transforming we will have a vector "At" referred to the cyclindrical basis: its orientation is represented by (ar, aφ, az).

That is A=f(x,y,z)*ax+g(x,y,z)*ay+h(x,y,z)*az ==> transform to ==> At=ft(r,φ,z)*ar+gt(r,φ,z)*aφ+ht(r,φ,z)*az

Where is the mismatch ?

Click to expand...

There is no mismatch so far. Although your calculation of the Jacobian is non-standard, but I understand.

We agree that z is not transformed; we can safely drop z from the current discussions.

Simple way is by an example: what will be the polar coordinates of a point whose Cartesian coordinates are given as (1,1)?

Use the answer from the book and you will see that the r is calculated correctly.

How do you get the correct value of the angle?