This is what I understand:
1 the cross is Cartesian product, The function is f(x,t) x is a vector in R^n and t is in [0,oo)
2 C^oo is a infinitely differentiable function with continuous derivatives.
I have been looking at the Navier Stokes equations and the formal presentation of them in the Millennium prize problems. There are two pieces of notation I cannot understand, I was hoping someone here could explain what they mean.
1.
I know R^{n} means n dimensional space and [0, infinity) is the range of values greater than or equal to zero and less than infinity but I do not know what the cross between them signifies.
The context this is in is "a<b on "
2.
The context this is in is
and where p and u are physical quantities.
I really don't know what this could mean but I'm sure it is not just a value raise to the power of infinity.
The document where this is all contain is here
http://www.claymath.org/millennium/N...vierstokes.pdf
P.s. I don't expect to solve these equations, I just have fun trying
Hi Shakarri!
The cross in between signifies a cartesian product.
It means that you construct a set with elements that are ordered pairs.
In your case these ordered pairs consist of a position vector and time.
The elements of are for instance ((0,0,0),0) or more generally .
The symbol signifies function that can be continuously differentiated infinitely many times.
In other words, it's a smooth function.
So no weird behavior with jumps or some such.
The function is of the type that will typically occur as which may for instance be the density of the fluid at location and at time t.
Divergence-free means that the divergence of the vector field is zero everywhere.
This translates to the fact that the density of the fluid remains constant - it is incompressible.
The amount of fluid that flows into a given volume will always be the same as the amount that flows out.