This was a "challenge" a professor gave us. I've

>>8383507
try using quantum mechanics

Def of D doesn't make sense to me.

>>8383561
agreed... to submit this to a *general* forum you also must define upper-case tau.

>>8383561
This is tensorial calculation, I've done that when I studied mechanics years ago.
grad = ( d/dx, d/dy, d/dz )
V = ( Vx, Vy, Vz )
grad V is the grad tensor represented by a matrix :
( d/dx V )
( d/dy V )
( d/dz V )
=
(d/dx Vx, d/dx Vy, d/dx Vz)
(d/dy Vx, d/dy Vy, d/dy Vz)
(d/dz Vx, d/dz Vy, d/dz Vz)

now OP has to tell us what he means with nabla x V
and then we can help

>>8383747
>now OP has to tell us what he means with nabla x V
The rotation, are you dumb?

>>8383784
don't be rude faggot
of course I know what a cross product is
but I'd rather had OP confirming his notations

>>8383747
Makes sense thanks. Ops problem seems pretty trivial then

OP here, the notation used is exactly what >>8383747 told, for cartesian coordinates. Nabla x V is the rotation, a cross product of the divergent vector by velocity.
I changed my approach and realized a few things. First, the tensor div(V) can be split into a symetrical part and a antissymetrical part, and the symmetrical part is D, defined on the original image.

div(V) = D+R

R=(1/2)*(div(V)-(div(V))^T)


this ^T means transpose

So I just have to show that the antisymmetrical part, R, have no effect on the product with the rotation w.
Any suggestions?

>>8384256
can't you just write out the components? too messy?

>>8384256
I don't know why you're introducing the divergence.

Plus I think you have the matrices / vectors the wrong way in your text.

You want to prove D.w = nabla v .w

D.w = (1/2) (nabla v . w + T nabla v . w)
= (1/2) (naba v . w) + (1/2) T nabla v . w
= (1/2) nabla v . w + (1/2) ( nabla (v.w) - nabla (w) . v)
= (1/2) nabla v . w - (1/2) nabla w . v
because v.w = 0

>>8384372
where did the transpose go?

>>8385073
it's been lost in translation

File: chanprob1.gif (9KB, 467x450px) Image search: [Google]

9KB, 467x450px

>>8384308
Here ya go OP.

>>8385102
that 2w near the end should be 4w, and ignore the spaces like "a z" in the last matrix. I think it works though. There must be an easier way I'm sure.