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. $\displaystyle R^n \times [0,\infty)$

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 $\displaystyle R^n \times [0,\infty)$"

2. $\displaystyle C^{\infty}$

The context this is in is http://puu.sh/2jdXP

and http://puu.sh/2je5B 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 :)

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.

Re: Help understanding this notation

Quote:

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. $\displaystyle R^n \times [0,\infty)$

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 $\displaystyle R^n \times [0,\infty)$"

2. $\displaystyle C^{\infty}$

The context this is in is

http://puu.sh/2jdXP
and

http://puu.sh/2je5B 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 $\displaystyle \mathbb R^n \times [0,\infty)$ are for instance ((0,0,0),0) or more generally $\displaystyle (\mathbf x, t)$.

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

Re: Help understanding this notation

Thank you very much, that is much clearer!