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- 12-23-2015, 05:28 PM #1

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## The application of the Mathematical definition of Newton's derivative to Physics

I begin the thread with a discussion of John Gabriel's misunderstanding of the Newton/Leibniz derivative, since it actually is a subtle point that underlies the coordinate analysis necessary to understand Theory of Relativity (both Special and General) as well as the foundation of Quantum Mechanics, especially in terms of affine vector spaces.

From the thread: Is 0.999.. = 1.0? (a brief digression from an otherwise totally useless thread...

A New Calculus

Ah, wonderful. He does know who he is......

John seems to be using the "mainstream" chain rule for higher order derivatives.

So mainstream calculus is ok after all?

That said, it is true that John has at long last defined the "New Calculus" derivative for a straight line in the (x,y) plane:

by his wonderful construction:

where, given A, the equation is true for ALL VALUES of x and y in the straight line. Not only that, but one can change the derivative simply by choosing different a's, b's, and C's.**One doesn't even have to refer to x in the definition of A**(the "New Derivative"); one can simply set the variation equal to zero by expanding in terms of (x+), and setting = 0 in the expansion), eliminating all terms except that defining the Ultimate Linear Line.

Of course, you can apply this formula to polynomials by simply taking "mainstream" consecutive derivatives until you arrive at the straight line, in which case you can then apply John's wonderful formula....

I repeat: this derivative is true for all straight lines in the (x,y) plane and applies at all values of x and y on the line.

So with this perspective for straight lines one doesn't even need Newton or Leibnitz's definitions of the derivative, and can perform all the physical and mathematical analyses on straight lines. Ergo, Newton's and Kepler's formulae for planetary motion must be misguided, and since there are no curves (we got rid of them at the second order level by successive differentiation), quantum mechanics and wave equations must be wrong as well.

Not only that, but the only real values are those where the numbers are integers, so the only possible slopes must be Pythagorean, and the only allowable numerical relations must be in terms of the lowest common denominator of these Pythagorean triples.

What a contribution to the world of mathematics! No wonder everyone is applauding John's .. Well, ok, someone must be somewhere. Well, ok, at least John is ....

All John is doing by moving his line (which he thinks is a "tangent" to an arbitrary function at arbitrary x) up and down is changing the value of C in the equation:

y = Ax + C

(where C is the value of the intercept of the line with the Y axis at x = 0.

The problem is that since A is not a related to f(x) at x, the equation is not a simultaneous solution of an arbitrary function f(x) and the line y = Ax + C, so while the line has the same slope (possibly, if there is a connection; there may not be if f(x) or x is piecewise), it only touches the line by inspection (imagination)....

For a LINE, then:

y = Ax

(with intercept at the origin, so C=0),

However, for an arbitrary function f(x), A must depend on x, so:

y(x) = f(x) = A(x)*x

Then f'(x) = f(x)/x = A(x) Where A(x) now depends on the position x.

Since this is true for all x, (wherever it is), A(x) depends only on the ratio f(x)/x, so we can specify the ratio at a particular position of x by letting an infinitesimal distance decrease along the abscissa (x), which will correspond to a similar infinitesimal variation in f(x).

Therefore, for any arbitrary position of x (as opposed to length from the origin), we can substitute:

,

where the differential means that we are considering the ratio f'(x) at a single point, rather than defining it relative to the origin. In fact, the differential means that we are taking the origin of the coordinate system to be at the point

(x,y) = (x,f(x)) (ignoring C)

and x doesn't even have to go to zero to define the tangent; the only requirement is that it be equal as a variation in both x and f(x).

Therefore, the derivative is just as well defined at x=0 as A = f'(x)= f(x)/(x), where x is arbitrary, since y+y=f'(x)(x+x) defines the line at (x,y)=(0,0):

i.e. f'(x,y) =f'(0,0) = f'(0)_{y=0 at x=0 }

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Changing the labels around, (so that x(0) and y(0) are the "original" coordinate system with the differential taken at (0',0') gives ....

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The point of taking the limit is that the variation is insignificant (so the space is "flat") compared to x (small enough so there is no curvature, or second order approximation at the point where the derivative is taken). The difficulty with John's analysis is that the "parallels" are different for each point on a curved line (i.e., the coordinate system "rotates" at different points on the curved line - suggesting a Lorentz transform - sort of-- )...

(John's "parallel tangents" is sort of analogous to parallel transport in differential geometry, GTR, but not exactly. One needs Newton and Einstein for that

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(hint:, one can think of A as a coordinate transformation in the equation x'=Ax, extended to two dimensions by y = Ax)Last edited by BuleriaChk; 12-24-2015 at 01:51 PM.

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"Flamenco Chuck" Keyser

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