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I have a tensor defined by $T_{ij} = \delta_{ij} + \epsilon_{ijk}x_k$, and I want to find $\int_ST_{ij}dS$, where S is the surface of the unit sphere.

I'm having some problems because I really don't know what integrating a tensor is. I'm also confused about what the surface integral is doing, since the dS is scalar not a vector (i.e. $ndS$ )

If I work in Cartesians I think $x_k$ is the normal to the sphere, but that has led to confusion because I think I should be working in sphericals.

Even integrating $\delta_{ij}$ has confused me, because of the scalar surface element.

Any help would be really really appreciated :)

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  • $\begingroup$ Can you also where you see this notation? $\endgroup$
    – user99914
    Sep 17, 2015 at 6:58
  • $\begingroup$ What do you mean? $\endgroup$
    – ThanksABundle
    Sep 17, 2015 at 7:00
  • $\begingroup$ Where did you see this notation (note, paper....?) $\endgroup$
    – user99914
    Sep 17, 2015 at 7:03
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    $\begingroup$ Well, from the notation I would guess that it is nothing but integrating each component of $T_{ij}$ term by term on $S$. $\endgroup$
    – user99914
    Sep 17, 2015 at 8:21
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    $\begingroup$ In general if you want to calculate $\int_S f(x) dS$, you have to find out a parametrization for $S$, (that is a mapping $F: U \to S$) and then do the calculation on $U$. But in this special case where your function is $x_i$ and $S$ is the unit sphere, you can use the fact that $x_i$ is an "odd" function on $S$, so the integral is zero. $\endgroup$
    – user99914
    Sep 17, 2015 at 8:29

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