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I'm sorry if this has been asked before and if the title is unclear; I'm not sure how to search for this question. Is it proper to write a function that is a function it's variables derivative as:

$$f(x)=\frac{dx}{dt}=\dot{x}$$

instead of

$$f(\dot{x})=\dot{x}$$

where $x=x(t)$? This (possibly) creates a contradiction in that

$$f(x)=f(\dot{x})$$

Thanks in advance!

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    $\begingroup$ I'm not exactly sure what is your question; people write $\dot{x} = f(x)$ informally to mean the differential equation $\dot{x}(t) = f(x(t))$ for all $t$ on some domain. $\endgroup$ – Qi Zhu Oct 25 '19 at 17:22
  • $\begingroup$ That's a good point. Here's an example. If I'm trying to define the force $f$ due to friction, I would say something like $f(\dot{x})=-k\dot{x}$. But I could also write $f(x)=-k\frac{dx}{dt}=-k\dot{x}$ couldn't I? $\endgroup$ – eball Oct 25 '19 at 18:29
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    $\begingroup$ Well, the two equations you've written are different. The first takes $\dot{x}$ as the parameter, the second $x$. I think you probably mean the second equation. (Yes, you can write the second equation, here $f$ is a function of functions $x$ and you apply the differential operator on the right which is in itself a function.) $\endgroup$ – Qi Zhu Oct 25 '19 at 22:12
  • $\begingroup$ @QiZhu, I see what you're saying. Would you agree that both $f(\dot{x})=-k\dot{x}$ and $f(x)=-k\frac{dx}{dt}$ are acceptable ways to write it, but maybe the second way is canonical? $\endgroup$ – eball Oct 28 '19 at 11:48
  • $\begingroup$ No, I‘m saying that these two mean different things. $\endgroup$ – Qi Zhu Oct 28 '19 at 12:51
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I cannot imagine a function that is a function of its own derivative. Sounds like a circular reference.

Perhaps you mean this. Consider a function $x = x(t)$ and you can have a function of both $x(t)$ and $x'(t)$ at the same time? For example, consider an iteration of Newton's method, which computes $$ N(t, x(t), x'(t)) = t - \frac{x(t)}{x'(t)}. $$ This is well-defined as a map $N : \mathbb{R}^3 \to \mathbb{R}$.

If you want to map functions to other functions, it's a different story. For example, consider the space $\mathcal{S}$ of all one-variable functions, differentiable infinitely many times on $[0,1]$, e.g. $\cos(x)$ or $e^x$ or any polynomial. Then you can have a map $D : \mathcal{S} \to \mathcal{S}$ defined by $$ D[f(x)] = f'(x), $$ thus mapping $e^x \to e^x$ and $\sin x \to \cos x$, for example.

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  • $\begingroup$ I think I meant "function that is a function of its variables derivative". Overall, I think what I'm asking is if it makes sense to write $f(x)=\dot{x}$. From your answer, it sounds like no, but I'm not quit sure why. $\endgroup$ – eball Oct 25 '19 at 17:11

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