How to find the curl of a vector field which points in the theta direction?

  • #26
Orodruin
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What I have found is that when we say "vector ##(r, \theta, \phi)##"
No. Unlearn this as fast as possible. We never say anything like this. You need to separate the notion of points from vectors. In a Euclidean space (or any other affine space) there is a bijection between the space itself and its tangent vector space. This is where the position vector comes in. However, in general, coordinates have little to do with vectors. ##(r,\theta,\phi)## is not a vector. It is a list of coordinates identifying a particular point. The position vector of this point is ##\vec x = r\hat r##, no component in the ##\hat\theta## or ##\hat\phi## directions (but note that ##\hat r## depends on both angular coordinates!)
 
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  • #27
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No. Unlearn this as fast as possible. We never say anything like this. You need to separate the notion of points from vectors. In a Euclidean space (or any other affine space) there is a bijection between the space itself and its tangent vector space. This is where the position vector comes in. However, in general, coordinates have little to do with vectors. ##(r,\theta,\phi)## is not a vector. It is a list of coordinates identifying a particular point. The position vector of this point is ##\vec x = r\hat r##, no component in the ##\hat\theta## or ##\hat\phi## directions (but note that ##\hat r## depends on both angular coordinates!)
I would really like to learn and know about that "bijection between the space itself and its tangent vector space". And thank you for making it clear that ##(r, \theta, \phi)## is a point and with respect to this point ##\vec x = r \hat r## is a position vector.
 
  • #28
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The confusion with the notation ##(x,y,z)## or ##(r,\theta,\phi)## for points, is that in many books it is also used to denote vectors for example the vector $$\vec{A}=A_x\hat x+A_y\hat y+A_z\hat z$$ can be denoted as $$(A_x,A_y,A_z)$$ and it is done so in many books.
And to add to this confusion, the position vector to the point (x,y,z) and in cartesian coordinates it is $$\vec{P}=x\hat x+y\hat y+z\hat z$$ or ##\vec{P}=(x,y,z)## if we use the alternate notation, so in cartesian coordinates the notation for the point ##(x,y,z)## and for the position vector to that point is identical. But in spherical coordinates it is not identical as i wrote in post #22.
 
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  • #29
benorin
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What can I say but that after vector calc, real analysis (both under grad and grad sequences) I thought I knew vectors. Nope.

Moderator: you may delete my posts in this thread as they are incorrect and I don't want to confuse ppl who happen upon this page.
 
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  • #30
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What can I say but that after vector calc, real analysis (both under grad and grad sequences) I thought I knew vectors. Nope.

Moderator: you may delete my posts in this thread as they are incorrect and I don't want to confuse ppl who happen upon this page.
One thing I know, it was you benorin whose solution on “volume enclosed by three intersecting cylinders” was understanble. It was your solution (I even did some private messaging with you about that) that helped me in understanding.
 
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  • #31
Delta2
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Well we all are humans, we can do mistakes. No one is perfect.

@benorin The fact that you admitted your mistake I believe it says a lot about your ethos.
 
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  • #32
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@Delta2 thanks. Bonus points for quoting he who was my hero during my romance with philosophy in college: Socrates.
 
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  • #33
etotheipi
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I think a Cartesian coordinate system is actually defined as an affine coordinate system with origin ##O## and an orthonormal basis {##\vec{e}_x, \vec{e}_y, \vec{e}_z##} where the coordinates of ##P## are a tuple containing the components of the position vector ##\overrightarrow{OP} = x\vec{e}_x + y\vec{e}_y + z\vec{e}_z##.

Part of the confusion might be because it is not obvious how a polar/spherical coordinate system is defined, because the basis vectors depend on the position. I.e. we can only say ##\overrightarrow{OP} = r\hat{r}##, and the components of this vector evidently don't give us the coordinates.

Perhaps someone (@Orodruin/@Delta2) might be able to elaborate on how the mapping from ##A## to ##\mathbb{R}_n## is achieved for polar coordinates?
 
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  • #34
Orodruin
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Part of the confusion might be because it is not obvious how a polar/spherical coordinate system is defined, because the basis vectors depend on the position. I.e. we can only say ##\overrightarrow{OP} = r\hat{r}##, and the components of this vector evidently don't give us the coordinates.
This is the key issue. In a Cartesian coordinate system, the coordinates also happen to be the components of the position vector. This is generally not true. A coordinate transformation is defined as a 1-to-1 map between two different sets of coordinates. It has no a priori relationship to the components of the position vector, but is just a function from an n-tuple of numbers to another n-tuple of numbers. For example, relating polar coordinates on ##\mathbb R^2## to Cartesian coordinates,
$$
x = \rho \cos\phi, \quad y = \rho \sin\phi.
$$
 
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