Understanding Double Inner Product Calculation in Multivariable Calculus

[Smed]

Hi, I'm having trouble understanding how to perform the following calculation:

$$u=(u,v,w)$$

$$(\nabla u + (\nabla u)^T) : \nabla u$$

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.

$$2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial v}{\partial y}\right)^2 + 2\left(\frac{\partial w}{\partial z}\right)^2$$

Could someone please shed some light on how the double inner product should work?
Thanks

 

[HallsofIvy]

Hi, I'm having trouble understanding how to perform the following calculation:

$$u=(u,v,w)$$

$$(\nabla u + (\nabla u)^T) : \nabla u$$

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 $\nabla u$ 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 $\cdot$?

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.

$$2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial v}{\partial y}\right)^2 + 2\left(\frac{\partial w}{\partial z}\right)^2$$

Could someone please shed some light on how the double inner product should work?
Thanks

 

[torquil]

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