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Math Help - Help understanding this notation

  1. #1
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    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
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  2. #2
    Member kezman's Avatar
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    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.
    Thanks from Shakarri
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  3. #3
    Super Member ILikeSerena's Avatar
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    Re: Help understanding this notation

    Quote Originally Posted by Shakarri View Post
    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.
    Thanks from Shakarri
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  4. #4
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    Re: Help understanding this notation

    Thank you very much, that is much clearer!
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