Can dp/dt Be Found When p(x) Is Inverse?

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Discussion Overview

The discussion revolves around the mathematical exploration of the derivative of a function \( p(x) \) when \( p \) is expressed in terms of an inverse function. Participants are investigating the relationship between \( \frac{dp}{dt} \), \( \frac{dp}{dx} \), and the limits involved in these derivatives, particularly in the context of parametric 2D vectors and their behavior as parameters approach certain values. The scope includes theoretical reasoning and mathematical derivation.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that evaluating \( \frac{\Delta p}{\Delta t} \) as \( \Delta t \) approaches zero could relate to \( \frac{dp}{dx} \) before reaching \( \frac{dp}{dt} \.
  • One participant suggests that an expression for \( t \) could be \( t = x^{-1}(dx) \) and questions if \( \frac{dp}{dt} \) can be found for that \( t \).
  • Another participant expresses uncertainty about obtaining an expression for \( \frac{dp(x^{-1}(dx))}{dt} \) that is evaluable at any \( t \).
  • There is a discussion about the formal definition of limits and derivatives, with one participant arguing that the notion of "approaching" in the context of limits does not imply a temporal process.
  • Some participants are exploring the behavior of derivatives of parametric 2D vectors and the conditions under which the derivative may have a direction normal to the curve.
  • Questions arise regarding the interpretation of derivatives in the context of vector functions and whether the zero-length point of a derivative can exist.
  • One participant mentions the calculus of variations in relation to differentiating with respect to a function.
  • There is a reference to rules about the derivative of an inverse function, although uncertainty remains about their application in the current context.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are multiple competing views regarding the interpretation of derivatives, the application of limits, and the behavior of parametric vectors. The discussion remains unresolved with ongoing questions and explorations.

Contextual Notes

Limitations include the dependence on definitions of derivatives and limits, as well as the unresolved nature of the mathematical steps involved in the proposed expressions.

rabbed
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If p is a function of x which is a function of t and you evaluate delta_p/delta_t as
delta_t goes to zero, it should be possible that delta_p/delta_t equals delta_p/dx
(or dp/dx) before reaching dp/dt.
Is it possible to find an expression for t where this happens?

Hm.. maybe when t = x^-1(dx) ?
Is it possible to find dp/dt for that t?
 
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$$\frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt}=
\lim_{\Delta t \to 0} \frac{p(t+\Delta t) - p(t)}{\Delta t}$$
 
I know, but that doesn't get me an expression of dp(x^-1(dx))/dt that can be evaluated at any t, does it?

On second thought, it should be dp(t+x^-1(dx))/dt

I'm trying to follow what happens to a 2D vector derivative when it starts to grow orthogonal to the tangent..
 
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Possible to turn this limit expression into derivatives?

Lim delta_x-> 0: ( p(x^-1(x+delta_x)) - p(x^-1(x)) ) / x^-1(delta_x)

Where x^-1(x) = t
 
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rabbed said:
If p is a function of x which is a function of t and you evaluate delta_p/delta_t as
delta_t goes to zero, it should be possible that delta_p/delta_t equals delta_p/dx
(or dp/dx) before reaching dp/dt.
You appear to be using the intuitive idea that a "limit" involves the notion of something "approaching" something else over an interval of time or in a step-by-step fashion. If you look at the formal definition of "##\lim_{t \rightarrow a} f(t)## you will find that the definition does not define any process taking place in time or in a sequence of steps. Since the definition of a derivative is based on the definition of limit, the definition of derivative also does not involve a process of something "approaching" something else as time passes or as a number of steps are executed. So your question doesn't have any defined meaning in mathematics, because there is no process described in the definition of derivative that would involve a "before" or "after".

If you are talking about algorithms to approximate derivatives, these often do involve a specific sequence of steps. But in order to determine if a variable in such an algorithm "reaches" a certain value before another value, you would have to say which particular algorithm you are asking about.
 
Hi Stephen

I'm trying to increase understanding of what happens to a parametric 2D vector when you take its derivative.
Letting two points of a curve approach each other by letting the parameter difference go to zero, there should
be a point where the derivative has a direction normal to the curve but has length 0, and then the length should
start to grow, still having the same direction.
I'm thinking that maybe the zero-length point occurs when delta_t = x^-1(dx), and as you decrease delta_t down
to dt the length starts to grow. It should make some sense, since the zero-length point should exist?
 
Wouldn't this take you closer to that idea?

Lim delta_x-> 0: ( p(x^-1(x+delta_x)) - p(x^-1(x)) ) / x^-1(delta_x)

Seems it's called calculus of variations, if you derivate wrt a function?
 
rabbed said:
Hi Stephen

I'm trying to increase understanding of what happens to a parametric 2D vector when you take its derivative.
You apparently are thinking of some algorithm or process to approximate the derivative because, as I mentioned, "taking" a derivative is not defined in terms of process that takes place in time or in a series of steps.

Your original post didn't mention a vector. Apparently you mean a function ##F(x) = (f_1(x), f_2(x))## whose domain is a set of real numbers and whose codomain is a set of two dimensional vectors?

Letting two points of a curve approach each other by letting the parameter difference go to zero, there should
be a point where the derivative
Which derivative? ##(f_1'(x), f_2'(x))##?

has a direction normal to the curve but has length 0

Why do you think that? Suppose the curve is ##F(x) = (f_1(x), f_2(x)) = (x, x+1)## with ##x(t) = t##. Where is there a point point on the curve where ##(f_1'(x),f_2'(x))## is normal to the curve?
 
Stephen Tashi said:
Your original post didn't mention a vector. Apparently you mean a function F(x)=(f1(x),f2(x))F(x) = (f_1(x), f_2(x)) whose domain is a set of real numbers and whose codomain is a set of two dimensional vectors?
I know, sorry. But this would apply to each partial derivative so I thought it would be simpler to discuss for just one variable.

Stephen Tashi said:
Why do you think that? Suppose the curve is F(x)=(f1(x),f2(x))=(x,x+1)F(x) = (f_1(x), f_2(x)) = (x, x+1) with x(t)=tx(t) = t. Where is there a point point on the curve where (f′1(x),f′2(x))(f_1'(x),f_2'(x)) is normal to the curve?
I'm thinking of a picture like the one in the answer here: http://math.stackexchange.com/quest...e-of-a-vector-orthogonal-to-the-vector-itself
delta_v would at some point become a zero vector, before starting to grow? And since the vector derivative is created by derivating each component, it should apply to a function of a function of a single variable also?
 
  • #11
Stephen Tashi said:
Then why don't you ask the question that the answerer answered?
Since I want to study this per component
 
  • #12
rabbed said:
Since I want to study this per component

Then it isn't clear what you are asking. If you can't find the words to express your general question, try asking about a specific example.
 
  • #13

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