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Tensor Integration

  1. Nov 13, 2009 #1
    Could someone help me out ??

    I tried this integration over the surface of a sphere of unit radii,

    P_{mn}e_{m}\otimes e_{n}=\frac{1}{D_{pq}e_{p}\otimes e_{q}}\int e_{m}\otimes e_{n}dS_{r=1}\][/tex]

    and I always get [tex]\[
    4\pi e_{m}\otimes e_{n}\][/tex] and the 'D' tensor as it is..

    I am expecting additionally a '3' in the denominator, am I wrong ??? If i do the integration over unit volume then I get the 3 in the denominator. Sorry for sounding stupid but is there a necessity to consider the unit tensor, i just assume it as a constant under integration.
  2. jcsd
  3. Nov 13, 2009 #2
    Homework assignments and stuff like that should be posted in the appropriate section.

    I don't get your indices either; it seems they don't add up on left and right hand sides. [tex] e_m \otimes e_n [/tex] definitely needs not be constant. Consider for example usual spherical coordinates.
  4. Nov 13, 2009 #3

    George Jones

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    Okay, I'll bite; along with clamtrox's note about indices, I have questions. What does

    [tex]\frac{1}{D_{pq}e_{p}\otimes e_{q}}[/tex]

    mean? How does one divide by a tensor (not the component of a tensor), which is an element of a vector space?
    Yes and no. From the Physics Forums rules:
  5. Nov 14, 2009 #4
    hey !!

    Thanks guys for looking at my work.

    I cant see how the indices dont add up... maybe i am missing something... but

    Each component of
    \mathbf{P}\] [/tex]will be a function of the [tex]\mathbf{\mathrm{D}^{-1}}[/tex] tensor.

    about division by the tensor..

    [tex]x=\mathbf{D}y[/tex] for some 'x' and some 'y'

    so I hope I can rewrite this as [tex]y=\mathbf{\mathrm{D}^{-1}}x[/tex]
    and probably find the Inverse at a later stage. Which for the time being I believe doesent depend on the co-ordinates of integration.

    Like clamtrox said, I use spherical co-ordinates to integrate, should I worry about [tex]\[
    e_{m}\otimes e_{n}\][/tex] should I transform the tensor basis ???
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