# Math Help - Help understanding this notation

1. ## Help understanding this notation

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. $R^n \times [0,\infty)$
I know Rn 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 $R^n \times [0,\infty)$"

2. $C^{\infty}$
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

2. ## Re: Help understanding this notation

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.

3. ## Re: Help understanding this notation

Originally Posted by Shakarri
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. $R^n \times [0,\infty)$
I know Rn 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 $R^n \times [0,\infty)$"

2. $C^{\infty}$
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 $\mathbb R^n \times [0,\infty)$ are for instance ((0,0,0),0) or more generally $(\mathbf x, t)$.

The symbol $C^\infty$ 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 $p$ is of the type $\mathbb R^n \times [0,\infty) \to \mathbb R$ that will typically occur as $p(\mathbf x, t)$ which may for instance be the density of the fluid at location $\mathbf x$ 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.

4. ## Re: Help understanding this notation

Thank you very much, that is much clearer!