http://mathworld.wolfram.com/

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Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.




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--Last edited by ferme on 2007-01-20 17:38:19 --

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ferme  ~

        DO you know understand linear al, and tensors

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--Last edited by saucer on 2007-01-22 16:52:49 --

A SCALAR is a TENSOR whose RANK is zero, and a VECTOR is a TENSOR whose RANK is one. You will need to see is what is meant by a tensor of rank 2, or 3 etc.  As you can see, TENSOR is an inclusive term, a generalization of the concept of
vector.
To illustrate a tensor of rank 2, imagine a plane surface area with a force acting on it. The total effect depends on two things, the magnitude and direction of the force, and the size of the area and its orientation. In fact this latter property can be represented uniquely by a vector of magnitude proportional to the size of the area, and in a direction NORMAL to the area. So the effect of the force upon the surface depends on two vectors and is called a TENSOR of RANK TWO.

In fact, if you consider components of force and each of these
components acting on each component of the area vector, there are nine terms in all, which can be displayed as an array, representing the total stress, and this quantity is the tensor of rank two. Tensors can thus be represented by arrays, and manipulated in a manner reminiscent of matrix manipulation. They have particular importance in problems involving invariance in changes of a coordinate system.

I suppose, this is a general definition for such.  

--Last edited by ferme on 2007-01-20 21:13:48 --

In general, if you are dealing with n-dimensional space, a tensor of
rank 2 has n^2 components.

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Math papers, resource



http://www.1millionpapers.com/



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The Maths Forum @ Drexel:


Can dy/dx Be Treated as a Fraction?

Date: 08/26/2004 at 14:57:25
From: Amit
Subject: Properties of dy/dx

When I learned about derivatives, I learned that dy/dx was a notation
that implied "derivative of y with respect to x."  I understood that.
But I am confused about whether or not the notation dy/dx can be
treated as a fraction, giving individual meanings to dy and dx.

For example, in integration by substitution:

integrate (sin(3x+5)dx)
 u = 3x+5
 du/dx = 3
 dx = du/3

That is the part that confuses me.  How can the dx be solved for?  
What exactly is "dx"?

I understand that dy/dx is a limit, and that it is a slope.  But the
idea of it being simply a notation doesn't help me understand how you
can multiply out the bottom.  Any help would be appreciated...


Date: 08/26/2004 at 15:27:40
From: Doctor Vogler
Subject: Re: Properties of dy/dx

Hi Amit,

Thanks for writing to Dr Math.  The easy answer to your question is
that your definition for dy/dx is correct; it means the derivative of
y with respect to x, and dy and dx are meaningless when written alone,
so that

 dx = du/3

is not a meaningful expression but should be written

 dx/du = 1/3.

And when certain nice things happen that *look* like fractions, such as:

           1
 dy/dz = -----
         dz/dy

and

 dz/dx = dz/dy * dy/dx

then this is actually just the Chain Rule at work.  And the reason that

 integral( f(g(x)) g'(x) dx ) = integral( f(u) du )

is not that

 u = g(x)

implies

 du = g'(x) dx

but rather the Chain Rule again.

All of that is true, except that I should qualify the "not a
meaningful expression."  You see, something is only meaningless until
somebody gives it a formal meaning.  Then you hope that the meaning
they gave it has useful properties (such as, that it relates to
derivatives...).  In fact, this has been done, and there is a good
deal of mathematics that has gone into the theory of differentials,
and it fits into integrals, and putting the differential "dx" at the
end of every integral also makes sense according to this theory, and
so on