Understanding Double Inner Product Calculation in Multivariable Calculus

In summary: 's answer is correct - you should use the dot product of the first term and the dot product of the second term.
  • #1
36
1
Hi, I'm having trouble understanding how to perform the following calculation:

[tex]
u=(u,v,w)
[/tex]

[tex]
(\nabla u + (\nabla u)^T) : \nabla u
[/tex]

I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

[tex]
2\left(\frac{\partial u}{\partial x}\right)^2
+ 2\left(\frac{\partial v}{\partial y}\right)^2
+ 2\left(\frac{\partial w}{\partial z}\right)^2
[/tex]

Could someone please shed some light on how the double inner product should work?
Thanks
 
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  • #2
Smed said:
Hi, I'm having trouble understanding how to perform the following calculation:

[tex]
u=(u,v,w)
[/tex]

[tex]
(\nabla u + (\nabla u)^T) : \nabla u
[/tex]
Your notation here doesn't make sense to me. If you are using "u" to represent a vector, don't use the same "u" to represent one of its components. If you are writing [itex]\nabla u[/itex] as, say, a row vector, then [itex[(\nabl u)^T[/itex] would be a column vector and you cannot add them. In general, a vector and its transpose are in different vector spaces and cannot be added. Finally, I don't know what ":" means. Was that supposed to be [itex]\cdot[/itex]?

I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

[tex]
2\left(\frac{\partial u}{\partial x}\right)^2
+ 2\left(\frac{\partial v}{\partial y}\right)^2
+ 2\left(\frac{\partial w}{\partial z}\right)^2
[/tex]

Could someone please shed some light on how the double inner product should work?
Thanks
 
  • #3
I think it is related to the definition in section 1.3.2 found here:
http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html

It is for a pair of rank 2 tensors, and is denoted by a :

The \nabla u's used in the original post are interpreted as second rank tensors, and the double inner product is applied between the terms on each side of the :

Haven't got time to check the calculation myself.

Torquil
 

1. What is a double inner product?

A double inner product is a mathematical operation that takes two vectors and produces a scalar value. It is similar to a regular inner product, but it involves two vectors instead of just one.

2. How is a double inner product calculated?

To calculate a double inner product, you first take the inner product of the two vectors individually. Then, you multiply those two inner products together to get the final scalar value.

3. What is the significance of a double inner product in mathematics?

A double inner product is important in mathematics because it allows us to measure the similarity between two vectors. It is also used in various mathematical concepts, such as orthogonal projections and least squares regression.

4. Can a double inner product be negative?

Yes, a double inner product can be negative. This can happen when the two inner products being multiplied together have opposite signs. It is important to pay attention to the signs when calculating a double inner product.

5. How is a double inner product used in real-world applications?

A double inner product has various real-world applications, such as in signal processing, image and video compression, and machine learning. It is also used in physics, particularly in quantum mechanics, to calculate the probability of measuring a certain state.

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