Anyone know the difference between dx (or dt) and deltax (or deltat)?

[Alan Grayson]

If you do, you'll also know why instantaneous frames of motion don't exist.

[Lawrence Crowell]

The “delta” ultimately has reference to a map from a vector space to scalars or “functions of functions” in mathematics such as the calculus of variations.  For a δx there is the meaning of x as a function with x = x(t), for instance, so that δx = (∂x/∂t)δt, and δt the parameter of variation. 

 

LC

[Roahn Wynar]

The best way to come to understand the difference is through the study of the Calculus of Variations. You don't have to study it too deeply either. Deltax is actually an arbitrary function which identifies the variation of a path.  dx actually is a bit ambiguous depending on context. It can either be understood as a form, dual basis vector, or the plain old measure of an integral. 

Roahn

Sent from my iPad

On Sep 5, 2013, at 4:47, Alan Grayson <agrays...@gmail.com> wrote:

If you do, you'll also know why instantaneous frames of motion don't exist.

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[Alan Grayson]

I disagree. Both dx and deltax appear first in standard calculus, way before the student ever heard of Calculus of Variations; the former after the limit is taken, the latter after -- but I agree that dx is ambiguous, and not merely a "bit". But more important in this discussion is my contention that instantaneous frames of reference don't exist, unless perhaps if one is willing to assume time is infinitely divisible, thereby introducing a form of infinity into physics.

[Philip Thrift]

    A dx (or a dt) goes with a ∫, a Δx (or a Δt) goes with a ∑.



- pt

[Lawrence Crowell]

The δ in calculus comes with the defintion of continuity and the derivative.  A function is continuous  if for the interval (x, x’) there exists a real number δ > 0 with |x – x’| < δ and |f(x) – f(x’)| < ε the function is continuous in the limit x --- > x’.

 

LC

[Alan Grayson]

On Sat, Sep 7, 2013 at 7:26 PM, Lawrence Crowell <goldenfield...@gmail.com> wrote:

The δ in calculus comes with the defintion of continuity and the derivative.  A function is continuous  if for the interval (x, x’) there exists a real number δ > 0 with |x – x’| < δ and |f(x) – f(x’)| < ε the function is continuous in the limit x --- > x’.

 

LC

True. I should have made it clear the delta I was referring to is associated with d, such as dx verses deltax. The former is not a real number, but an indication that the limit has been taken, say in calculating a derivative.  But more important, do you agree there are no instantaneous frames of reference, such as a frame existing for time dt?

On Friday, September 6, 2013 10:43:22 PM UTC-5, Alan Grayson wrote:

I disagree. Both dx and deltax appear first in standard calculus, way before the student ever heard of Calculus of Variations; the former after the limit is taken, the latter after -- but I agree that dx is ambiguous, and not merely a "bit". But more important in this discussion is my contention that instantaneous frames of reference don't exist, unless perhaps if one is willing to assume time is infinitely divisible, thereby introducing a form of infinity into physics.

On Fri, Sep 6, 2013 at 7:53 PM, Roahn Wynar <rwy...@comcast.net> wrote:

The best way to come to understand the difference is through the study of the Calculus of Variations. You don't have to study it too deeply either. Deltax is actually an arbitrary function which identifies the variation of a path.  dx actually is a bit ambiguous depending on context. It can either be understood as a form, dual basis vector, or the plain old measure of an integral. 

Roahn

--

[Alan Grayson]

On Sat, Sep 7, 2013 at 11:22 PM, Alan Grayson <agrays...@gmail.com> wrote:

On Sat, Sep 7, 2013 at 7:26 PM, Lawrence Crowell <goldenfield...@gmail.com> wrote:

The δ in calculus comes with the defintion of continuity and the derivative.  A function is continuous  if for the interval (x, x’) there exists a real number δ > 0 with |x – x’| < δ and |f(x) – f(x’)| < ε the function is continuous in the limit x --- > x’.

 

LC

True. I should have made it clear the delta I was referring to is associated with d, such as dx verses deltax. The former is not a real number, but an indication that the limit has been taken, say in calculating a derivative.  But more important, do you agree there are no instantaneous frames of reference, such as a frame existing for time dt?

In retrospect, I did state at the outset that I was asking about deltax or deltat, as compared to dx or dt.