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Homework Help: Covariant & Contravariant Components

  1. Sep 17, 2015 #1
    1. The problem statement, all variables and given/known data
    This is really 3 questions in one but I figure it can be grouped together:

    1. The vector A = i xy + j (2y-z2) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis.

    2. Transform the vector in Problem 1 into its covariant components in spherical coordinates.

    3. Transform the vector in Problem 1 into its contravariant components in spherical coordinates.

    2. Relevant equations

    Transform equations, relationship to between covariant and contravariant components.

    3. The attempt at a solution

    I feel really bad because this is like the 3rd time I have posted on here in an attempt to get help but I am out of options with my college professor. Last time I asked for help his only response was STUDY HARDER! It is really hard to when he doesn't provide decent examples and solid theory.

    Anyways, I need help understanding some of what he is talking about above. I know how to transform A into spherical coordinates. does the unit vector basis referred to mean in r,θ,φ? If so I know what he is asking then and can solve that. My REAL problem, is figuring out how to come up with covariant and contravariant components. Will they end up being vectors? I have checked wikipedia and several questions on this forum but just can't seem to make the connection.

    ANY help at all is greatly appreciated.
  2. jcsd
  3. Sep 17, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    A good place to start is by reading on-line sources; for example, the article
    offers some pretty nice explanations.
  4. Sep 18, 2015 #3
    Thanks for the reply. I have already read that page and still cannot fully grasp the concept. I am having a lot of difficulty relating these concepts together.
  5. Sep 18, 2015 #4
    I think I can help you.

    You say you can solve problem 1. Let's see what you got.

    Are you familiar with the term "coordinate basis vectors." If so, please define what they mean.

  6. Sep 18, 2015 #5
    Ok I will get back to you in a little bit I am at work. I will workout that first part. Thanks for the replies!
  7. Sep 21, 2015 #6
    Ok been a crazy few days. Haven't had a chance to work the first part. The coordinate basis vectors refer to the unit vectors right? So in this case he wants the vector transformed in spherical coordinates with respect to the r,θ,φ right? I feel like every since I started this class my understanding of vectors has regressed.
  8. Sep 21, 2015 #7
    No. The coordinate basis vectors are defined for spherical coordinates by:
    where ##\vec{r}## is a position vector from the origin. You need express your vector A in terms of the three coordinate basis vectors in order to determine its contravariant components. Now, how are the coordinate basis vectors related to the unit vectors in the r, θ, and φ directions?
    Yes, for part 1 of your problem.

  9. Sep 21, 2015 #8
    Ok. I have attached my solution to problem 3. I ended up with some pretty long terms.

    Attached Files:

  10. Sep 22, 2015 #9
    Your methodology is correct, but not your algebra. I found at least 1 error: in one of the j terms, you lost an exponent of 2 on r in going from the first equation to the second equation. There may be other algebraic errors. Please check it over.

    Also, I can't read your handwriting with regard to distinguishing between the φ's and the θ's. I'm wondering whether you can.

    I don't want to proceed further until you're sure that your algebra is correct.

  11. Sep 22, 2015 #10
    Ok I attempted to write bigger and used Φ instead of φ to hopefully make it more visibly different. Again, thank you for all the help so far!

    Attached Files:

  12. Sep 22, 2015 #11
    I haven't checked your "arithmetic," but that's not really necessary for what we want to do next. To get us started on the contravariant components, I need you to try to answer the question I asked in post #7.

  13. Sep 27, 2015 #12
    Ok. So is there some more transformation stuff I need to do or do I simply need to take what I have already transformed and group stuff together differently?
  14. Sep 27, 2015 #13
    For an orthogonal coordinate system like this, it is just a matter of grouping stuff differently (particularly in getting the contravariant components). The key to this lies in post #7.

  15. Sep 27, 2015 #14
    Ok cool. I will work this some more and try to have another post by tomorrow. Thanks!
  16. Oct 7, 2015 #15
    How's this? Sorry been busy trying to buy a house and with work.

    Attached Files:

  17. Oct 7, 2015 #16
    Sorry, no. It's much simpler than that. In terms of the unit vectors:
    Now you have two equations for dr, one in terms of the unit vectors and the other in terms of the coordinate basis vectors. So, how are the coordinate basis vectors related to the unit vectors?

  18. Oct 8, 2015 #17
    Would that be a relation of the unit vector dr=∂r/∂x*Ar?
  19. Oct 8, 2015 #18
    No. The relationships would be
    $$\vec{a}_r=\frac{\partial \vec{r}}{\partial r}=\vec{i}_r$$
    $$\vec{a}_θ=\frac{\partial \vec{r}}{\partial θ}=r\vec{i}_θ$$
    $$\vec{a}_φ=\frac{\partial \vec{r}}{\partial φ}=r\sinθ\vec{i}_φ$$
    Can you see by comparing the equations in posts #16 and #7 where these equations come from?

  20. Oct 8, 2015 #19
    Oh yes I can now! Sorry I didn't realize I was comparing those two. But after you said it I realize I was being dumb!
  21. Oct 9, 2015 #20
    Suppose I have an arbitrary vector ##\vec{V}##. I can express it in component form in terms of the spherical coordinate unit vectors by writing:
    where the parentheses in the subscripts signify that I am talking about the components with respect to the unit vectors. I can also represent the exact same vector ##\vec{V}## in terms of the coordinate basis vectors by writing:
    The superscripted components in this equation are called "the contravariant components of the vector ##\vec{V}##." Given the relationships between the unit vectors and the coordinate basis vectors that I presented in post #18, how are the contravariant components related to the components expressed in terms of the unit vectors?

  22. Oct 12, 2015 #21
    So by using what you have given me from the previous post I have:


    or in terms of the components Vr=V(r)/1 ; Vθ=V(θ)/r ; Vφ=V(φ)/rsinθ
  23. Oct 12, 2015 #22
    Good job. These are the contravariant components of the vector.

    Now for the covariant components. To get the covariant components, we need to introduce another set of basis vectors. These are called the "reciprocal basis vectors," and are defined as follows:
    $$\vec{a}^r\centerdot \vec{a}_r=1$$
    $$\vec{a}^θ\centerdot \vec{a}_θ=1$$
    $$\vec{a}^φ\centerdot \vec{a}_φ=1$$
    The dot product of ##\vec{a}^r## with the other two coordinate basis vectors is zero. The same goes for ##\vec{a}^θ## and ##\vec{a}^φ##.

    We are going to express the vector ##\vec{V}## in terms of the reciprocal basis vectors and the covariant (subscripted) components of ##\vec{V}## as follows:

    So tell me what the reciprocal basis vectors are in terms of the unit vectors and also in terms of the coordinate basis vectors.

    (Note that the analysis we have done so far is strictly for an orthogonal coordinate system, in this case, spherical coordinates). For a non-orthogonal coordinate system, the relationships are similar, but a little more complicated.)
  24. Oct 19, 2015 #23
    Would I have: ar= 1 / ir aθ= 1 / iθ*r aφ= 1 / iφ* rsinθ

    or in other words. Divide Vr,θ,φ by their scale factors?
  25. Oct 20, 2015 #24
    No. In vector analysis, we never have a vector appearing in the denominator of any equation (unless it's dotted there with something else). The three reciprocal basis vectors are:

    ##\vec{a}^r=\vec{i}_r##, ##\vec{a}^θ=\frac{\vec{i}_θ}{r}##, and ##\vec{a}^φ=\frac{\vec{i}_φ}{r\sinθ}##

    Tell me if this makes sense.

  26. Oct 20, 2015 #25
    Yes it does. So now that I know what these relationships are I need to apply them to the transformation I solved for correct?
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