Hello! Sorry I just saw this reply. As an answer, I got (rcos(θ)sin(θ)er +rcos2(θ)eθ +(2cos(θ)sin(θ)z+ 2sin(θ)cos(θ)z)ezCharles Link said:The unit vector ## \hat{i} ## also needs to be put in cylindrical coordinates. ## \hat{i}=cos(\theta) \hat{a}_r -sin(\theta) \hat{a}_{\theta} ## if I computed it correctly. Once you do that, you should get agreement when you do the curl operation in cylindrical coordinates. ## \\ ## Editing: Yes, I computed it, and got agreement. I'd be happy to check the answer for you that you get.
Please check your ## e_z ## term. The first term gets a minus sign so that the ## e_z ## term is zero. Also what did you get in Cartesian coordinates for the curl? The two should agree and I think they do if the ## e_z ## term is zero.Bestphysics112 said:Hello! Sorry I just saw this reply. As an answer, I got (rcos(θ)sin(θ)er +rcos2(θ)eθ +(2cos(θ)sin(θ)z+ 2sin(θ)cos(θ)z)ez
Is this answer correct?
Edit- Yes! I figured out my mistake and I got an equivalent answer. Thank you for the help Charles
I don't have my homework with me, but I forgot to edit my post with the answer I got. The ez terms canceledCharles Link said:Please check your ## e_z ## term. The first term gets a minus sign so that the ## e_z ## term is zero. Also what did you get in Cartesian coordinates for the curl? The two should agree and I think they do if the ## e_z ## term is zero.
Recasting a vector field in cylindrical coordinates allows for a more intuitive understanding and visualization of the field. It also simplifies calculations and makes it easier to solve certain problems.
To convert a vector field from Cartesian coordinates to cylindrical coordinates, you need to use the following equations:
r = √(x2 + y2)
θ = tan-1(y/x)
z = z
Once you have these values, you can rewrite the vector field in terms of r, θ, and z.
Cylindrical coordinates are especially useful when dealing with cylindrical objects or problems with cylindrical symmetry. They allow for simpler and more efficient calculations and can often provide a more intuitive understanding of the vector field.
Yes, any vector field can be recast in cylindrical coordinates as long as the equations and transformations are applied correctly. However, it may not always be the most practical or useful approach depending on the problem at hand.
One limitation of using cylindrical coordinates is that they are not suitable for all types of vector fields. For example, if the field has a lot of z-component variation, it may be better to keep it in Cartesian coordinates. Additionally, the equations and transformations for cylindrical coordinates can become more complex for higher dimensions, making it more difficult to recast the vector field.