Switching coordinate system of a field

In summary: It decomposes the change in ##A_x## into two components, one coming from each of the two changes.In summary, the conversation discussed transforming a vector field in cylindrical coordinates to Cartesian coordinates. The equations for this transformation were provided, and the meaning of certain terms was clarified. The conversation also addressed how to find the change in A_x with time, which involves both the change in A_r and the change in the angle θ.
  • #1
2sin54
109
1

Homework Statement


Say I have some sort of a vector field in the cylindrical coordinate system [tex] \vec{F}(r, \Theta, z) = f(\vec{A}(r,\Theta,z),\vec{B}(r,\Theta,z)) [/tex]

How do I switch to the Cartesian coordinates? More precisely, how do I transform [tex] A_r = g(A_x,A_y,A_z), A_\Theta = h(A_x,A_y,A_z)[/tex] and so on?

Homework Equations



https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates

The Attempt at a Solution



I understand that [tex]{\left(A_z\right)}_{cyl.} = {\left(A_z\right)}_{Cart.} [/tex]
and similarly for B and that
[tex] A_x = A_r\cdot\cos(\Theta), A_y = A_r\cdot\sin(\Theta [/tex].

However, what do I do with [tex] A_\Theta[/tex]?
 
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  • #2
##\vec{A}=A_r~\hat{r}+A_{\theta}~\hat{\theta}##
##\hat{r}=\cos \theta~\hat{x}+\sin \theta~\hat{y}## and ##\hat{\theta}=-\sin \theta~\hat{x}+\cos \theta~\hat{y}##
Therefore,
##\vec{A}=A_r~(\cos \theta~\hat{x}+\sin \theta~\hat{y})+A_{\theta}~(-\sin \theta~\hat{x}+\cos \theta~\hat{y})##
From which
##A_x = A_r~\cos \theta - A_{\theta}~\sin \theta~;~~ A_y = A_r~\sin \theta+A_{\theta}\cos \theta##
Is this what you are looking for?
 
  • #3
kuruman said:
##\vec{A}=A_r~\hat{r}+A_{\theta}~\hat{\theta}##
##\hat{r}=\cos \theta~\hat{x}+\sin \theta~\hat{y}## and ##\hat{\theta}=-\sin \theta~\hat{x}+\cos \theta~\hat{y}##
Therefore,
##\vec{A}=A_r~(\cos \theta~\hat{x}+\sin \theta~\hat{y})+A_{\theta}~(-\sin \theta~\hat{x}+\cos \theta~\hat{y})##
From which
##A_x = A_r~\cos \theta - A_{\theta}~\sin \theta~;~~ A_y = A_r~\sin \theta+A_{\theta}\cos \theta##
Is this what you are looking for?
I don't see [tex]A_z[/tex] in your equations. Why is that? Also, one can express [tex]\Theta[/tex] as [tex] = \arctan(\frac{y}{x})[/tex] correct? But what's the meaning of it when x = 0? If I am solving a physics problem then surely the set of points for which x = 0 aren't necessarily impossible.
 
  • #4
You don't see Az because it is the same in Cartesian and cylindrical coordinates. I omitted it to save time, but you can assume it's there.
2sin54 said:
But what's the meaning of it when x = 0? If I am solving a physics problem then surely the set of points for which x = 0 aren't necessarily impossible.
What makes you think these points might be impossible? When x = 0, the vector points either along the +y axis in which case θ = π/2 or along the -y axis in which case θ = 3π/2.
 
  • #5
Thank you for your answers. One more question. Assuming [tex]A_x = A_r\cdot\cos(\Theta), A_\Theta = 0[/tex] If I wish to find how Ax changes with time is the following approach correct?
[tex] \frac{dA_x}{dt} = \frac{dA_r}{dt}\cdot\cos(\Theta) - A_r\cdot\sin(\Theta)\cdot\frac{d\Theta}{dt}[/tex]
 
  • #6
That is correct. It says that when ##A_x## changes with time, it can happen either because ##A_r## changes or because the angle changes or both.
 

Related to Switching coordinate system of a field

1. What is a coordinate system?

A coordinate system is a set of rules and conventions used to assign numerical values to a location in space. It allows us to measure and describe the position, orientation, and movement of objects or phenomena in a systematic way.

2. Why would you want to switch coordinate systems of a field?

Switching coordinate systems of a field can be useful for various reasons. It can help simplify complex mathematical equations or make it easier to visualize and analyze data. It may also be necessary to align the coordinate system with a different reference point or to integrate data from different sources.

3. How do you switch coordinate systems of a field?

The process of switching coordinate systems of a field involves transforming the coordinates from one system to another using mathematical equations. This can be done manually, but it is often more efficient to use specialized software or programming techniques to perform the transformation.

4. What are some common coordinate systems used in fields?

Some common coordinate systems used in fields include Cartesian coordinates, polar coordinates, and spherical coordinates. Each system has its own set of rules and conventions, and the choice of which system to use depends on the specific application and needs of the scientist.

5. Are there any challenges or limitations when switching coordinate systems of a field?

Yes, there can be challenges and limitations when switching coordinate systems of a field. The transformation process can be complex and may require advanced mathematical knowledge. Additionally, some systems may not be compatible with others, making it difficult to accurately transform the coordinates. It is important to carefully consider the potential limitations and accuracy of the transformation when switching coordinate systems.

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