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The cylindrical coordinate system was first introduced to me in a multivariable calc class, though now it's being used again in a different context. When dealing with various fancy integrals, we were mostly concerned with the change of variables from x, y, and z to rho, phi, and z. Conversion was fairly straightforward, since it is easily seen how x = rhocosphi and y = rhosinphi and dxdydz becomes rhodrhodphidz. However, I don't understand the unit vector side of the definition of the cylindrical coordinate system very well.
When simply dealing with the cylindrical system to describe a position, phi is defined as the angle counterclockwise from the x-axis to rho. In the vector definition, howerver, rho, phi, and z form an orthonormal basis. I don't understand this for two reasons--how is it possible to relate phi as an angle giving the position of rho projected in the x-y plane with phi as an unit vector? Does it make sense to have phi be used as a direction when rho in the original definition has both an x and y component?
TL DR-> When converting a vector in the standard basis to a cylindrical basis, why do I need to convert both the variables x, y, z and ALSO use the corresponding transformations mandated by the change from the unit vectors ax, ay, az to arho, aphi, az.
 
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Sorry, it's been too long since I did this shit. Sounds exciting though. Makes me want to go study it again. I forgot how cool this stuff is. Don't you have a textbook to consult? Or a professor?
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United States10328 Posts
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical.
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On October 05 2012 16:35 ]343[ wrote:
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical.
Pretty sure it's r dr dphi dz; r sin phi is for spherical polar coords.
I'm not really sure what you're asking either. I find http://mathworld.wolfram.com/CylindricalCoordinates.html to be a pretty good collection of information (seriously that page is packed) on cylindrical coords.
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On October 05 2012 16:42 Oldfool wrote:Show nested quote +On October 05 2012 16:35 ]343[ wrote:
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical. Pretty sure it's r dr dphi dz; r sin phi is for spherical polar coords. I'm not really sure what you're asking either. I find http://mathworld.wolfram.com/CylindricalCoordinates.html to be a pretty good collection of information (seriously that page is packed) on cylindrical coords.
That doesn't really answer my quesiton; I want to know the definitions that lead to (20), (21), and (22), as well as the relationship (if any) with (4), (5), and (6).
On October 05 2012 16:35 ]343[ wrote:
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical.
I think you've mixed spherical and cylindrical together with that.
I guess my actual question is the change of basis matrix i.e. given a vector x in the standard basis and say b = Ax where A is the matrix that changes x from the standard basis to the cylindrical basis vector b, how is the matrix A related to the original x = rhocosphi, y = rhosinphi, and z=z?
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On October 05 2012 17:46 Loser777 wrote:Show nested quote +On October 05 2012 16:42 Oldfool wrote:On October 05 2012 16:35 ]343[ wrote:
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical. Pretty sure it's r dr dphi dz; r sin phi is for spherical polar coords. I'm not really sure what you're asking either. I find http://mathworld.wolfram.com/CylindricalCoordinates.html to be a pretty good collection of information (seriously that page is packed) on cylindrical coords. That doesn't really answer my quesiton; I want to know the definitions that lead to (20), (21), and (22), as well as the relationship (if any) with (4), (5), and (6).
The relationship is that you take your new positional vector r and derive it with respect to its variables (rho, phi and z). This gives you an orthogonal system of vectors. To normalize you divide by their respective length, now you have the orthonormal vectors e_phi, e_rho, e_z.
Show nested quote +On October 05 2012 16:35 ]343[ wrote:
Sorry, I'm not sure I understand your question.
The dx dy dz = r sin phi dr dphi dz is a change of variables for the integral; the "unit vector" thing you're talking about sounds like a change of coordinate basis. These are related, but I don't think they're identical. I think you've mixed spherical and cylindrical together with that. I guess my actual question is the change of basis matrix i.e. given a vector x in the standard basis and say b = Ax where A is the matrix that changes x from the standard basis to the cylindrical basis vector b, how is the matrix A related to the original x = rhocosphi, y = rhosinphi, and z=z?
Which matrix relates the volume elements to one another? It's the Jacobian! Same here.
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On October 05 2012 17:46 Loser777 wrote:
I guess my actual question is the change of basis matrix i.e. given a vector x in the standard basis and say b = Ax where A is the matrix that changes x from the standard basis to the cylindrical basis vector b, how is the matrix A related to the original x = rhocosphi, y = rhosinphi, and z=z? Something to keep in mind is that the change from standard coordinates to cylindrical coordinates is not a linear transformation, so there isn't really a "change of basis matrix".
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how is it possible to relate phi as an angle giving the position of rho projected in the x-y plane with phi as an unit vector?
I simplified this to as "how is it possible to relate an angle giving a position to an unit vector?" because it doesn't matter what angle or which position (phi and rho).
Wouldn't you relate angles to values through the trigonometric functions?
So when x=rcos(theta) and y=rsin(theta) for your vector r = <x,y,z>, aka x = rhocosphi and y = rhosinphi, and for your vector r = <rho,phi,z>, you're taking a derivative with respect to phi (in the case of the phi unit vector) and then normalizing.
In the case of an actual vector, those phis will have values and thus give you your directional unit vector.
Does it make sense to have phi be used as a direction when rho in the original definition has both an x and y component?
I think so, because in the expression y = rhosinphi, what you're doing is multiplying rho by some number, given by sinphi. In the end, y will still equal a normal integer. Just because the value phi ITSELF doesn't technically have an 'x' or 'y' component because it's an angle doesn't mean it can't relate to them.
Edit: Well after reading your replies to the other comments I don't even know if what I said means anything (not that it was necessarily correct anyways, it's been a few months since multivariable). I think you're asking for a couple different things here. Yo no comprende.
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Aotearoa39261 Posts
I was going to write something up, but then I realised this and this do a pretty good job of explaining this.
e_rho always points along a line from the origin outwards
e_phi tells you, effectively, what is the angle that e_rho is pointing in - but it isn't as simple as that. It actually tells you how 'fast' the point/object is rotating around the origin
e_z tells you the height
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Might be easier to understand the same concept for polar coordinates (2D) first. For polar coordinates you have a radial unit vector which is just r/|r| and a tangential unit vector which is perpendicular to it and of unit length.
The cylindrical case is exactly the same, just add a unit vector in the z direction.
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I love reading these high-level math discussions, though my geometry knowledge isn't near enough to contribute in any meaningful way.
I am adding rhodrhodphidz to my list of band names, though.
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On October 05 2012 17:56 phiinix wrote:
...basic?
hahahaha.
Well I think this one was more easily and consistently answered than those order of operation problems that half of TL can't do ^^
On October 05 2012 20:29 AmericanUmlaut wrote:
I love reading these high-level math discussions, though my geometry knowledge isn't near enough to contribute in any meaningful way.
I am adding rhodrhodphidz to my list of band names, though.
Yeah... multivariable calculus is ~Calc 3. How would you pronounce it as a band name? rod-rod-fidz? Instead of "rho-d-rho-d-phi-d-z"?
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just think about it.
my calc 2 prof was an interesting guy. probably the best part of the course was observing how manipulating the z-axis in polar coordinate system created spirals from circles (where z was sorta like time). there was a lot of insight to be garnered about various transformations because the guy had an intuitive understanding of how various representations interacted.
where math is concerned, just try to keep things consistent. usually you're thinking in a way that's too complex and sort of overshooting the real issue. when you're approaching all these transformations, the only danger is that of inconsistency. just keep in mind that all these transformations are dealing with the same basic information, and try to keep your focus on the big picture; the point is to appreciate how it all coalesces. don't get lost in nuance.
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Something to keep in mind is that most of the time the notation is quite sloppy. i just skimmed through plexas first link and it seemed right there. You have to distinguish between the unit vector e_phi and the length of this vector |phi|. so while most of the time one writes "phi", it is meant like |phi| * e_phi. (and no proper distinguishing between phi and |phi|)
You should also keep in mind that the transformation between x,y,z < - > r, phi, z is different from dx dy dz because in the first you simply change the coordinates while at the latter you account the fact that a change in coordinates also changes the way volume is measured.
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This is not basic. Also Plexa...
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First, I want to say that I think it's completely reasonable that you were confused, and possibly good. There's a subtle distinction here and the chances are that your teacher did not mention.
You already understand the first part -- the transformation between Cartesian and cylindrical coordinates given by
x = r cos phi
y = r sin phi
z = z
Now you're looking at integrals, and then dx dy dz is becoming r dr dphi dz, why is that?
To answer this you should think about what you are doing when computing an integral. You split your space up into a bunch of tiny boxes. Then for each box you're evaluating your function in the middle of the box and multiplying by the size of the box. The integral is just the limiting value of this process.
Okay so consider what happens to this sum after a coordinate transformation. The value of the function won't change, but the size of the box might. Equally sized boxes in one coordinate system might not be equally sized in the other! The measure of how the size of the box changes is given by the Jacobian of the coordinate change.
You should be familiar with the idea that any smooth function, when you look at it closely enough, behaves pretty much as if it's linear. One can think of the Jacobian as just this linear map. But wait! The change of coordinates is not actually linear, what gives? Well what you actually want is a way to think about "local behavior" as another space. This is called the tangent space at a point, which I'll write as T_x for the tangent space at x. Now in R^3, the tangent space at every point also looks like R^3. Even more confusingly, in Cartesian coordinates, for every point x, the basis vectors dx, dy, dz of T_x just look like the normal basis vectors <1,0,0>, <0,1,0>, <0,0,1>.
But let's look at cylindrical coordinates. Given a point x = (r cos phi, r sin phi, z), what should the basis vectors of T_x be? They should describe the effect of increasing r by a small amount, increasing phi by a small amount, or increasing z by a small amount. What you'll get is that the basis vectors are dr = <cos phi, sin phi, 0>, dphi = <-r sin phi, r cos phi, 0>, dz = <0, 0, 1>. This is different for each point x, but will be an orthogonal basis of T_x for whatever x you choose (there are complications at the origin, but when doing integrals you don't care if one point is messed up).
The change in the size of a box is given by the determinant of the matrix defined by dr, dphi, dz, which you can compute to be r. That's why dx dy dz becomes r dr dphi dz.
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On October 06 2012 00:33 Hamster1800 wrote:
First, I want to say that I think it's completely reasonable that you were confused, and possibly good. There's a subtle distinction here and the chances are that your teacher did not mention.
You already understand the first part -- the transformation between Cartesian and cylindrical coordinates given by
x = r cos phi
y = r sin phi
z = z
Now you're looking at integrals, and then dx dy dz is becoming r dr dphi dz, why is that?
To answer this you should think about what you are doing when computing an integral. You split your space up into a bunch of tiny boxes. Then for each box you're evaluating your function in the middle of the box and multiplying by the size of the box. The integral is just the limiting value of this process.
Okay so consider what happens to this sum after a coordinate transformation. The value of the function won't change, but the size of the box might. Equally sized boxes in one coordinate system might not be equally sized in the other! The measure of how the size of the box changes is given by the Jacobian of the coordinate change.
You should be familiar with the idea that any smooth function, when you look at it closely enough, behaves pretty much as if it's linear. One can think of the Jacobian as just this linear map. But wait! The change of coordinates is not actually linear, what gives? Well what you actually want is a way to think about "local behavior" as another space. This is called the tangent space at a point, which I'll write as T_x for the tangent space at x. Now in R^3, the tangent space at every point also looks like R^3. Even more confusingly, in Cartesian coordinates, for every point x, the basis vectors dx, dy, dz of T_x just look like the normal basis vectors <1,0,0>, <0,1,0>, <0,0,1>.
But let's look at cylindrical coordinates. Given a point x = (r cos phi, r sin phi, z), what should the basis vectors of T_x be? They should describe the effect of increasing r by a small amount, increasing phi by a small amount, or increasing z by a small amount. What you'll get is that the basis vectors are dr = <cos phi, sin phi, 0>, dphi = <-r sin phi, r cos phi, 0>, dz = <0, 0, 1>. This is different for each point x, but will be an orthogonal basis of T_x for whatever x you choose (there are complications at the origin, but when doing integrals you don't care if one point is messed up).
The change in the size of a box is given by the determinant of the matrix defined by dr, dphi, dz, which you can compute to be r. That's why dx dy dz becomes r dr dphi dz.
This guy is awesome.
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On October 06 2012 00:50 stoned_rabbit wrote:Show nested quote +On October 06 2012 00:33 Hamster1800 wrote:
First, I want to say that I think it's completely reasonable that you were confused, and possibly good. There's a subtle distinction here and the chances are that your teacher did not mention.
You already understand the first part -- the transformation between Cartesian and cylindrical coordinates given by
x = r cos phi
y = r sin phi
z = z
Now you're looking at integrals, and then dx dy dz is becoming r dr dphi dz, why is that?
To answer this you should think about what you are doing when computing an integral. You split your space up into a bunch of tiny boxes. Then for each box you're evaluating your function in the middle of the box and multiplying by the size of the box. The integral is just the limiting value of this process.
Okay so consider what happens to this sum after a coordinate transformation. The value of the function won't change, but the size of the box might. Equally sized boxes in one coordinate system might not be equally sized in the other! The measure of how the size of the box changes is given by the Jacobian of the coordinate change.
You should be familiar with the idea that any smooth function, when you look at it closely enough, behaves pretty much as if it's linear. One can think of the Jacobian as just this linear map. But wait! The change of coordinates is not actually linear, what gives? Well what you actually want is a way to think about "local behavior" as another space. This is called the tangent space at a point, which I'll write as T_x for the tangent space at x. Now in R^3, the tangent space at every point also looks like R^3. Even more confusingly, in Cartesian coordinates, for every point x, the basis vectors dx, dy, dz of T_x just look like the normal basis vectors <1,0,0>, <0,1,0>, <0,0,1>.
But let's look at cylindrical coordinates. Given a point x = (r cos phi, r sin phi, z), what should the basis vectors of T_x be? They should describe the effect of increasing r by a small amount, increasing phi by a small amount, or increasing z by a small amount. What you'll get is that the basis vectors are dr = <cos phi, sin phi, 0>, dphi = <-r sin phi, r cos phi, 0>, dz = <0, 0, 1>. This is different for each point x, but will be an orthogonal basis of T_x for whatever x you choose (there are complications at the origin, but when doing integrals you don't care if one point is messed up).
The change in the size of a box is given by the determinant of the matrix defined by dr, dphi, dz, which you can compute to be r. That's why dx dy dz becomes r dr dphi dz. This guy is awesome.
Sometimes I wish I was that good at math:p
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On October 05 2012 15:56 Loser777 wrote:
The cylindrical coordinate system was first introduced to me in a multivariable calc class, though now it's being used again in a different context. When dealing with various fancy integrals, we were mostly concerned with the change of variables from x, y, and z to rho, phi, and z. Conversion was fairly straightforward, since it is easily seen how x = rhocosphi and y = rhosinphi and dxdydz becomes rhodrhodphidz. However, I don't understand the unit vector side of the definition of the cylindrical coordinate system very well.
When simply dealing with the cylindrical system to describe a position, phi is defined as the angle counterclockwise from the x-axis to rho. In the vector definition, howerver, rho, phi, and z form an orthonormal basis. I don't understand this for two reasons--how is it possible to relate phi as an angle giving the position of rho projected in the x-y plane with phi as an unit vector? Does it make sense to have phi be used as a direction when rho in the original definition has both an x and y component?
TL DR-> When converting a vector in the standard basis to a cylindrical basis, why do I need to convert both the variables x, y, z and ALSO use the corresponding transformations mandated by the change from the unit vectors ax, ay, az to arho, aphi, az.
It's easier (actually more definition than anything else) to think about the unit vector as <dz, drho, dphi>. That should help clear things up (because those actually HAVE a direction, assuming you've set where 0 and 2pi are).
Also, be careful. This is a common bone between mathematicians and physicists, this convention of where phi and rho should be.
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EPT
