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Orthogonal coordinate systems

  1. Jan 15, 2014 #1

    ShayanJ

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    Recently I've been studying about orthogonal coordinate systems and vector operations in different coordinate systems.In my studies,I realized there are some inconsistencies between different sources which I can't resolve.
    For example in Arfken,it is said that the determinant definition of the cross product is reserved after changing to a coordinate system other than Cartesian and also Foundations of the Electromagnetic theory by Reitz and Milford uses the same way for calculating a cross product in spherical coordinates as in the Cartesian system.
    But in documents I found on the internet,the cross product in other coordinate systems involves the [itex] h_i(=\sqrt{g_{ii}}) [/itex] coefficients.One example is here.
    But again,the calculations presented in some other documents(like this and this) are that much different in(maybe)notation that I have problem relating them.
    I wanna ask,is there a book which is considered as the ultimate reference that its notations and definitions are most widely used?
     
  2. jcsd
  3. Jan 15, 2014 #2
    I don't have Arfken, so I can't address this, but the development in Reitz and Milford that you describe makes sense to me.
    This looks correct to me.
    I like this write up quite a bit. This is close to the way I learned the subject. It shouldn't be hard to derive the cross product in the wiki write up from this.

    I think you have pretty much all you need. I don't know why Arfken does it differently.

    Chet
     
  4. Jan 15, 2014 #3

    ShayanJ

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    Here's what Arfken says in the "Mathematical methods for physicists" [Arfken and Weber,international edition,sixth edition]
    http://www.updata.ir/images/avfyzb0n8m6ev0f9lp6.jpg [Broken]
    Its removed in the seventh edition anyway!

    About the calculation in Reitz and Milford(which is done the same way in "Introduction to electrodynamics" by D.J.Griffiths)
    When calculating the power radiated by an oscillating dipole,it writes the surface integral of the poynting vector as [itex]\frac{1}{\mu} R^2\int_0^\pi E_{\theta} B_{\phi} 2\pi \sin{\theta} d\theta[/itex] (which are given in spherical coordinates with B having only [itex] \varphi [/itex] component non-zero and E having only [itex]\varphi[/itex] component zero!) but if we wanted to consider the h coefficients(The way wikipedia does it),it should have been [itex]\frac{1}{\mu} R^4 \int_0^\pi E_{\theta} B_{\phi} 2\pi \sin^2{\theta} d\theta [/itex]!!!(The point is all the calculations spoil up by changing the power of R no matter decreasing or increasing it so it seems this ignoring of h parameters has some experimental backup!!!)
    Yeah,I was thinking I can relate them by just spending enough time on it so my main problem is the calculation in Reitz and Milford and also the statement in Arfken!
     
    Last edited by a moderator: May 6, 2017
  5. Jan 15, 2014 #4
    You need to check to determine precisely what they are doing. In orthogonal coordinate systems, they can work with (1) the contravariant components of the vectors, (2) the coordinate basis vectors, (3) unit vectors in the coordinate directions, and (4) the components of the vectors expressed with respect to the unit vectors. They can even be working with combinations of these. You need to be sure you understand which combinations of these they are using. In spherical coordinates for example, there can be additional factors of r and factors of sinθ, depending on which combinations they use.

    Chet
     
  6. Jan 16, 2014 #5

    ShayanJ

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    You're right. So I checked it. If you take a look at the parts where it is calculating electric and magnetic fields from scalar and vector potentials,you can see that its using the formulas for gradient and curl that are usually given(which aren't usually accompanied by explanations about co-variance and contra-variance)so its the fourth option,the usual one!
    So the problem is not solved!
     
  7. Jan 16, 2014 #6
    This can't be that complicated. I'm not very familiar with electromagnetics, but I'd be willing to give it a try. Please write down in vectorial form what you are trying to calculate. It looks like you are trying the take the cross product of the E vector with the B vector, dot it with a differential area, and integrate it over the surface of a sphere. If we can state the problem correctly, we can try it in the three different ways (notations) and see what we get.

    Chet
     
  8. Jan 18, 2014 #7

    ShayanJ

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    Yes,that's the thing we're going to to do,And here's how Milford does it:
    He first calculates vector and scalar potentials as:
    [itex] \vec{A}=\frac{\mu_0}{4\pi}\frac{I_0 l}{r}\sin{[\omega(t-\frac{r}{c})]} (\cos{\theta}\hat{r}-\sin{\theta}\hat{\theta}) [/itex]
    [itex]
    \phi=\frac{l}{4\pi\varepsilon_0}\frac{\cos{\theta}}{r}[\frac{q(t-\frac{r}{c})}{r}+\frac{I(t-\frac{r}{c})}{c}]
    [/itex]
    Then he calculates electric and magnetic fields from the potenials.The magnetic field has only the [itex] \varphi [/itex] component and the electric field has r [itex] \theta [/itex] components.
    We want to have the surface integral of the Poynting vector taken on the surface of a sphere whose radius is going to approach infinity.Because the normal to the sphere is in the direction of [itex] \hat{r} [/itex],we only need the r component of the Poynting vector which is constructed by [itex] E_{\theta} [/itex] and [itex] B_\varphi [/itex] which are:
    [itex]
    B_\varphi=\frac{1}{r}\frac{\partial }{\partial r} (rA_\theta)-\frac{1}{r}\frac{\partial A_r}{\partial \theta}=\frac{\mu}{4\pi}\frac{I_0l}{r} \sin{\theta} [\frac{\omega}{c}\cos{[\omega(t-\frac{r}{c})]}+\frac{1}{r} \sin{[\omega(t-\frac{r}{c})] }]
    [/itex]
    [itex]
    E_\theta=-\frac{1}{r}\frac{\partial \phi}{\partial \theta}-\frac{\partial A_\theta}{\partial t}=
    -\frac{ lI_0\sin{\theta} }{4\pi \varepsilon_0}[( \frac{1}{\omega r^3}-\frac{\omega}{rc^2} ) \cos{[\omega(t-\frac{r}{c})]} -\frac{1}{cr^2} \sin{[\omega(t-\frac{r}{c})] }]]
    [/itex]
    The point is,in my idea, the r component of the cross product should be [itex] r^2\sin{\theta} E_\theta B_\varphi [/itex](or involve metric coefficients somehow!) but Milford says it is just [itex] E_\theta B_\varphi [/itex]
     
  9. Jan 18, 2014 #8
    OK. I think I can see now what's happening here. He is calling [itex] E_\theta [/itex] and [itex] B_\varphi [/itex] the "physical components" of these vectors (referenced to the unit vectors in the θ and [itex] \phi [/itex] coordinate directions), and you are calling [itex] E_\theta [/itex] and [itex] B_\varphi [/itex] the contravariant components of these vectors (referenced to the coordinate basis vectors in the θ and [itex] \phi [/itex] coordinate directions). So the two of you are using the same symbols to represent two different (but related) things. But, you should be using superscripts for the contravariant components of these vectors, not subscripts. You can also do everything using the covariant components, but then you need to be very careful keeping track of subscripts because, unfortunately, the physical components are also expressed in terms of subscripts (and they are not the same as the covariant components).

    I think that this is a correct assessment. I hope this helps.

    Chet
     
  10. Jan 18, 2014 #9

    ShayanJ

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    I'm a little confused!
    Could you explain more clearly or introduce a resource which explains it clearly?
     
  11. Jan 18, 2014 #10
    I thought that one of the resources you presented in your first post was pretty good:
    http://homepages.engineering.auckla...ectors_Tensors_16_Curvilinear_Coordinates.pdf

    Try going through that.
    Meanwhile, I'm going to try to present a summery of how some of this plays out for Spherical Coordinates. I hope this works for you.

    Differential Position Vector in terms of unit vectors and physical components of differential position vector:
    [tex]d\vec{r}=\vec{i_r}dr+r\vec{i_θ}dθ+rsinθ\vec{i_{\phi}}d\phi[/tex]
    Differential Position Vector in terms of coordinate basis vectors and contravariant components of differential position vector:
    [tex]d\vec{r}=\vec{a_r}dr+\vec{a_θ}dθ+\vec{a_{\phi}}d\phi[/tex]
    If we compare these two equations, we see that the coordinate basis vectors are related to the unit vectors by:
    [tex]\vec{a_r}=\vec{i_r}[/tex]
    [tex]\vec{a_θ}=r\vec{i_θ}[/tex]
    [tex]\vec{a_{\phi}}=rsinθ\vec{i_{\phi}}[/tex]
    There are a third set of basis vectors that are also used. These are the reciprocal bases vectors (also sometimes called basis 1 forms). These are designed such that:
    [tex]\vec{a^r}\centerdot \vec{a_r}=1[/tex]
    [tex]\vec{a^θ}\centerdot \vec{a_θ}=1[/tex]
    [tex]\vec{a^{\phi}}\centerdot \vec{a_{\phi}}=1[/tex]
    So, in terms of the unit vectors:
    [tex]\vec{a^r}=\vec{i_r}[/tex]
    [tex]\vec{a^θ}=\frac{\vec{i_θ}}{r}[/tex]
    [tex]\vec{a^{\phi}}=\frac{\vec{i_{\phi}}}{rsinθ}[/tex]
    An arbitrary vector [itex]\vec{V}[/itex] can be expressed in component form in three equivalent ways: In terms of the unit vectors, in terms of the coordinate basis vectors, and in terms of the reciprocal basis vectors:
    [tex]\vec{V}=\bar{V_r}\vec{i_r}+\bar{V_θ}\vec{i_θ}+\bar{V_{\phi}}
    \vec{i_{\phi}}=V^r\vec{a_r}+V^θ\vec{a_θ}+V^{\phi}
    \vec{a_{\phi}}=V_r\vec{a^r}+V_θ\vec{a^θ}+V_{\phi}
    \vec{a^{\phi}}[/tex]
    where the components with the overbars (which multiply the unit vectors) are referred to as the physical components, the components with the superscripts (which multiply the coordinate basis vectors) are referred to as the contravariant components, and the components with the subscripts (which multiply the reciprocal basis vectors) are referred to at the covariant components. These equations, together with relationships between the coordinate basis vectors and the unit vectors, or the relationships between the reciprocal basis vectors and the unit vectors can be used to express the physical components, the contravariant components, and the covariant components in terms of one another.

    If the cross product of the electric field intensity vector and the magnetic field intensity vector is expressed in terms of its physical components by [itex]\bar{E_θ}\bar{B_{\phi}}\vec{i_r}[/itex], then in terms of its contravariant components, it is given by [itex]E^θB^{\phi}r^2sinθ\vec{a_r}[/itex]
     
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