# Is it reasonable and common to interpret $dt$ as a time point (a point in time)? [duplicate]

I heard some one talked about the instantaneous and average velocities.

He was using $$\Delta t$$ to denote a time frame, $$dt$$ denote a time point.

average velocities $$\bar{v} = \dfrac{\Delta s}{\Delta t}$$

the $$\Delta t$$ part is indeed common. my concern is about the $$dt$$ part

wiki use the notation

instantaneous velocities $$v = \dfrac{ds}{dt}$$

Is it reasonable and common to interpret this way?

• Possible duplicates: physics.stackexchange.com/q/65724/2451 and links therein. – Qmechanic Aug 26 at 11:02
• I think this is off-topic: it's a question about how differentials are defined, and that's a question about analysis, not about physics, and would therefore be better asked on math SE, especially as physicists are generally fairly vague on this. – tfb Aug 26 at 11:22
• – Kyle Kanos Aug 27 at 11:59

$$\text dt$$ is not a point in time. It is an infinitesimal time interval. Physically, you could think of it as a time interval that is much much smaller than the relevant time scale of the system. Mathematically, it is the limit of $$\Delta t$$ as it approaches $$0$$ (not equal to $$0$$).
This just comes from the definition of the limit: $$v=\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x(t)}{\Delta t}$$
Limits are not the same thing as equality. Plugging in $$\Delta t=0$$ makes the above definition undefined.
• Thanks for your answer. Would you please give a concrete example about the relevant time scale of the system, such as a car, plane, rocket? – whnlp Aug 26 at 11:21
• Simply saying "it's undefined" doesn't really answer the question. Why is it undefined? Well, Suppose I say $x=0/0$, and I ask you to solve for $x$. What I'm asking is for the unique $x$ such that $0x=0$. It should be obvious that there is no unique $x$. But if $f(c)=0$ and $g(c)=0$, and I ask for the limit of $f(x)/g(x)$ as $x$ approaches $c$, then there often is a unique answer. – Solomon Slow Aug 26 at 11:39
• @SolomonSlow The question isn't "why is $c/0$ undefined". It's not the reason why $\text dt$ isn't $0$. It's just something to point out. It might not answer something you want answered, but that isn't the OP's question. If the OP wanted to know why we can't divide by $0$ then I would explain that. – Aaron Stevens Aug 26 at 12:01