Surface Integration of vector tensor product

In summary, the conversation is about a person struggling with an integral involving two spheres, and they are looking for help with integration by parts. They provide details about the variables involved and mention using a formula for the product of two functions. They also mention a link that might be helpful for solving the problem.
  • #1
praban
13
0
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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  • #2
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:
 

Related to Surface Integration of vector tensor product

1. What is surface integration of vector tensor product?

Surface integration of vector tensor product is a mathematical operation that involves calculating the integral of a vector-valued function over a given surface. It is commonly used in physics and engineering to calculate quantities such as flux and work done by a vector field on a surface.

2. How is surface integration of vector tensor product different from regular integration?

The main difference between surface integration of vector tensor product and regular integration is the domain of integration. In regular integration, the integral is calculated over a one-dimensional interval, while in surface integration, the integral is calculated over a two-dimensional surface.

3. What is the significance of surface integration of vector tensor product in physics?

Surface integration of vector tensor product is significant in physics because it allows us to calculate important physical quantities, such as flux and work, which are essential for understanding and predicting the behavior of physical systems.

4. What are some applications of surface integration of vector tensor product?

Surface integration of vector tensor product has many applications in different fields, including fluid mechanics, electromagnetism, and solid mechanics. It is used to calculate fluid flow rates, electric and magnetic flux, and stress and strain distributions in solid materials, among others.

5. What are some techniques for calculating surface integration of vector tensor product?

There are several techniques for calculating surface integration of vector tensor product, including the divergence theorem, Stokes' theorem, and Green's theorem. These theorems allow us to convert a surface integral into a more manageable line or volume integral, which can then be solved using traditional integration methods.

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