Surface Integration of vector tensor product

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SUMMARY

The discussion centers on the integration of a vector and tensor product over two spheres, specifically the integral ∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ, where ui represents the fluid velocity around sphere i, Tj is a tensor associated with sphere j, and nj is the unit normal on sphere j's surface. The user, Praban, seeks assistance with applying integration by parts to this integral, questioning the applicability of the formula for the product of two functions in this context. The conversation emphasizes the need for clarity on the integration by parts formula to proceed with the calculation.

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  • Understanding of vector calculus, specifically surface integrals.
  • Familiarity with tensor analysis and tensor notation.
  • Knowledge of integration techniques, particularly integration by parts.
  • Basic concepts of fluid dynamics, especially related to velocity fields.
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  • Review the application of integration by parts in vector calculus.
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Mathematicians, physicists, and engineers working on fluid dynamics and tensor analysis, particularly those dealing with complex integrals involving multiple dimensions and surfaces.

praban
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Hello,

It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij|

∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ

The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th sphere)
and Tj (p+rij) is a tensor over the j th sphere. nj is the unit normal on the surface of jth sphere.

I am thinking of doing it by integration by parts. But I am not sure if I can use the same formula for product of two functions in this case as well. Can someone help me? If I can write the correct formula for integration by parts, the rest I should be able to do.

thanks,
Praban
 
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praban said:
Hello,

It may be trivial to many of you, but I am struggling with the following integral involving two spheres i and j separated by a distance mod |rij|

∫ ui (ρ).[Tj (ρ+rij) . nj] d2ρ

The integration is over sphere j. ui is a vector (actually velocity of the fluid around i th sphere)
and Tj (p+rij) is a tensor over the j th sphere. nj is the unit normal on the surface of jth sphere.

I am thinking of doing it by integration by parts. But I am not sure if I can use the same formula for product of two functions in this case as well. Can someone help me? If I can write the correct formula for integration by parts, the rest I should be able to do.

thanks,
Praban
This link might be helpful. :wink:
 

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