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Proving covariant component is physical component times scaling factor

  1. Aug 10, 2011 #1
    1. The problem statement, all variables and given/known data


    The problem is from Mathematical Methods in the Physical Sciences, 3rd Ed. Ch10, Sec. 10, Q4. My question is a bit subtle as I have actually figured out the problem, just that I don't understand my solution. The problem reads:

    4) What are the physical components of the gradient in polar coordinates? [See (9.1)]. The partial derivatives in (10.5) are the covariant components of [itex]\nabla u[/itex]. What relation do you deduce between physical and covariant components? Answer the same questions for spherical coordinates, and for an orthogonal coordinate system with scale factors [itex]h_{1}, h_{2}, h_{3}[/itex].


    2. Relevant equations

    And we are given:

    (9.1) [itex]\nabla u= \hat{e}_{r}\frac{\partial u}{\partial r} + \hat{e}_{\theta}\frac{1}{r}\frac{\partial u}{\partial \theta} + \hat{e}_{z}\frac{\partial u}{\partial z} [/itex]

    (10.5) [itex]\frac{\partial u}{\partial x'_{i}} = \frac{\partial u}{\partial x_{j}}\frac{\partial x_{j}}{\partial x'_{i}} = \frac{\partial x_{j}}{\partial x'_{i}}\frac{\partial u}{\partial x_{j}}[/itex]

    as well as the definition given in the book of a covariant vector:

    (*) [itex]V'_{i} = \frac{\partial x_{j}}{\partial x'_{i}} V_{j}[/itex].

    and

    (**) [itex]\nabla u = \sum_{i=1}^{3} \hat{e}_{i} \frac{1}{h_{i}} \frac{\partial u}{\partial x_{i}}.[/itex]


    3. The attempt at a solution

    I found that the relationship in these situations is that the covariant components of [itex]\nabla u[/itex] are the physical components multiplied by the scaling factors. This is seen by looking at Eq. (**), for [itex]\hat{e}_{i}[/itex], the physical component is [itex]\frac{1}{h_{i}} \frac{\partial u}{\partial x_{i}}[/itex] and the covariant component is [itex]\frac{\partial u}{\partial x_{i}}[/itex].

    My question is: Does this relationship hold for any vector and not just gradients of scalar fields? How do we know? For example, I could define a vector V to be

    [tex]\vec V = \hat{e}_{r}\frac{\partial u}{\partial r} + \hat{e}_{\theta}\frac{\partial u}{\partial \theta} + \hat{e}_{\phi}\frac{1}{r sin(\theta)}\frac{\partial u}{\partial \phi}[/tex].

    Notice the middle term on right hand side has no scaling factor (this is not a gradient, I just made it up). This vector seems not to obey the relation between the physical components and covariant components.


    Thanks!
     
  2. jcsd
  3. Aug 10, 2011 #2
    Sounds similar to me, I never understand my solutions; monkey see, monkey do.
     
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