Can anyone check this identity please?

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In summary, the conversation discusses the validity of the identity VV = V\otimes V in relation to second order tensors. The speaker mentions attempting to prove it but facing difficulties with the components of the tensor and the nabla. Another speaker clarifies that the identity is not true unless Div(v) = 0, which aligns with their work in fluid mechanics. The topic of incompressible flow is also briefly mentioned, along with the speaker's current work on the vorticity equation in Bird's transport phenomena, specifically in relation to the Navier-Stokes equation and the use of the Levi-Civita symbol.
  • #1
Deathcrush
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is this identity true?

V is a vector, so VV is a second order tensor

I have tried to prove this but the components of the tensor appear always as operands of the nabla.

Thanks!

Div(VV)=v.(Grad(V))
 
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  • #2
The way to go about proving this is to write [itex]V\otimes V=V_{i}V_{j}[/itex] and write the divergence as [itex]\partial^{i}(V_{i}V_{j})[/itex]
 
  • #3
I know now, thank you. Actually that "identity" es not true unless Div(v)=0, because another term is missing. However that fits perfectly since I am working with fluid mechanics.
Thanks again!
 
  • #4
Incompressible flow?
 
  • #5
yes, actually, I was trying to take the curl of the Navier-Stokes equation, to get the vorticity equation, a problem in Bird's transport phenomena, the book asks you to deduce it in two forms, the first one is the one I have already, the second one involves the Levi-Civita symbol and I'm currently working on it :)
 

FAQ: Can anyone check this identity please?

1. What does it mean to "check an identity" in science?

Checking an identity in science typically refers to verifying a mathematical or scientific equation or statement. This involves ensuring that both sides of the equation or statement are equal, often through various mathematical operations or experiments.

2. Why is it important to have someone else check an identity?

Having someone else check an identity is important because it helps to eliminate potential errors or biases. This is especially crucial in scientific research, where accuracy and objectivity are essential.

3. How can I check an identity on my own?

You can check an identity on your own by using various mathematical techniques, such as algebraic manipulation, substitution, or solving for variables. Additionally, you can use experimental data or simulations to verify the identity.

4. What should I do if the identity I am checking does not hold true?

If the identity you are checking does not hold true, it is important to carefully review your calculations and methods. It may also be helpful to consult with other scientists or experts in the field to identify and correct any errors that may have been made.

5. Can a computer program be used to check an identity?

Yes, a computer program can be used to check an identity. In fact, many scientific and mathematical software programs have built-in functions for checking equations and statements. However, it is important to validate the results and ensure that the program is accurately performing the necessary calculations.

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