Surface Integration of vector tensor product

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The discussion centers on the challenge of integrating a specific expression involving two spheres, focusing on the integral of a vector field and a tensor over the surface of one sphere. The user is considering using integration by parts but is uncertain about applying the formula for the product of two functions in this context. Clarification on the correct integration by parts formula is sought to proceed with the calculation. Participants are encouraged to provide insights or references that could assist in resolving this integral problem. The conversation highlights the complexities of vector and tensor integration in fluid dynamics.
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:
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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