# Different expressions for distance & displacement : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r|, and |$$\int$$d$$\vec r|$

I came across these expressions in my book. And the book says that all these are different from each other. The expressions are : $$\intd|\vec r|$$, $$\int|d\vec r$$|, and $$|\intd\vec r|$$

Are they different from each other?

I know that

• $$|\intd\vec r|$$ means magnitude of displacement,

• $$\int|d\vec r$$| means total distance,

• But what about $$\intd|\vec r|$$?

I think it should mean total distance too, but I’m not sure if $$\intd|\vec r|$$ and $$\int|d\vec r$$| have the same meaning, do they?

Edit : Some of the answers say that the question is not very clear, and that a little more explanation would help. I’m not sure what else to add, so I’m attaching a picture of that page

• The name of this book? Feb 16, 2022 at 18:59

In the above Figure-01 we see the quantities $$\;\mathrm d\mathbf{r}, \Vert\mathrm d\mathbf{r}\Vert,\mathrm d\Vert\mathbf{r}\Vert\;$$ for an infinitesimal displacement from point $$\;\boldsymbol{1}\;$$ to point $$\;\boldsymbol{2}$$.

Note that

1. $$\mathrm d\mathbf{r}\;$$ is an infinitesimal vector
2. $$\Vert\mathrm d\mathbf{r}\Vert\;$$ is an always non-negative infinitesimal scalar representing the magnitude of $$\;\mathrm d\mathbf{r}$$, and
3. $$\mathrm d\Vert\mathbf{r}\Vert\;$$ is a real infinitesimal scalar representing the change of the magnitude of the position vector of point $$\;\boldsymbol{1}\;$$ to point $$\;\boldsymbol{2}$$. So, it's positive if increasing, negative if decreasing and zero if constant.

Let see now the integrals of these quantities along a curve $$\;C\;$$ as in Figure-02.

$$$$\Delta \mathbf{r} \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\!C\mathrm d \mathbf{r}\boldsymbol{=} \mathbf{r}_2\boldsymbol{- } \mathbf{r}_1 \tag{01}\label{01}$$$$ $$$$\Delta\Vert \mathbf{r}\Vert \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\! C\mathrm d \Vert \mathbf{r}\Vert \boldsymbol{=} \Vert \mathbf{r}_2\Vert\boldsymbol{-} \Vert \mathbf{r}_1\Vert \tag{02}\label{02}$$$$

So, these quantities depend on the start and end points $$\;\boldsymbol{1},\boldsymbol{2}\;$$ and are independent of the curve joining them. I don't think that these quantities are useful in any case.

Now, $$$$s_{1}^{2} \boldsymbol{=}\int\limits_{1}^{2}\!\!\!\!\!\! C\Vert \mathrm d \mathbf{r}\Vert \boldsymbol{=}\text{lenght of curve between points }\boldsymbol{1},\boldsymbol{2} \tag{03}\label{03}$$$$

Note :

This textbook is confusing. On one hand it's written that we have a vector $$$$\vec s \boldsymbol{=}\Delta \vec r \boldsymbol{=}\lim_{\Delta \vec r_{i}\boldsymbol{\rightarrow} 0} \sum\limits_{\boldsymbol{i=1}}^{\boldsymbol{i=n}}\Delta \vec r_{i}\boldsymbol{=} \int d\vec r \tag{A}\label{A}$$$$ and then this same integral of the rhs is used for a scalar, the length of the line joining points 1,2 $$$$\int d\vec r\boldsymbol{=}\text{length of the line joining point 1 to 2}\boldsymbol{=}\vert \Delta \vec r\vert \tag{B}\label{B}$$$$

• The two diagrams that you have used for the explanation make the difference between these three notations very clear. Thanks for taking the time to answer the question. So $\int d|\vec r|$ is basically a scalar, and is equal to the difference between the magnitudes of $\vec r_1$ and $\vec r_2$ But does it have a physical meaning?
– 4d_
Feb 6, 2019 at 11:01
• @π times e : I don't think that $\int d|\vec r|$ has any physical meaning. Feb 6, 2019 at 12:26

Looks like an integral over radial distance.

• This probably could use more exposition to be a useful answer. Feb 3, 2019 at 12:23
• I'd edit my question and attach a picture to make it clearer
– 4d_
Feb 3, 2019 at 13:21

It is the radial part of the total distance along some path. For example, for a circle there is an angular part of the total distance ($$2\pi r$$), but there is no any radial part contribution: $$dr=0$$.

• Is this radial path (or radial distance) part of 'circular motion' ? I haven't studied circular motion yet, currently I'm staying motion in one dimension & motion in a plane. I'm not sure why it would show up in the book in the chapter I am studying. Do you think this expression can mean something else? Something related to what I am studying right now?
– 4d_
Feb 3, 2019 at 13:20
• A motion in a plane may be circular: a point makes a two dimensional path around some point staying always at a certain distance $r=R$ from it. Feb 3, 2019 at 18:51
• My bad! Yeah circular motion is an example of 2D motion too. I meant to say that, I'm not studying circular motion right now. It is the next chapter. But this thing showed up in the current chapter that I am studying
– 4d_
Feb 3, 2019 at 19:00

Notation matters. You have probably seen $$\int d|\vec{r}|$$ written as $$\int dr$$, without vectors.

• In 1D, this is the same as $$\int dx$$. The letter for the disttance is not relevant.
• In more than 1 dimension, it's the integral along the radius. For example, in a circle, you'd integrate for all angles and also for all radii from 0 to $$R$$, for example. There you'd find that integral. $$S=\int_0^{2\pi}d\varphi \int_0^R dr$$
• I just attached the pic, it’s exactly the same notation as I have used in my question. Now that I’ve attached the pic, can you please have another look at it? Thanks
– 4d_
Feb 3, 2019 at 14:26
• Your picture is helpful to understand the two former cases, but you ask for a third case, $d|\vec{r}|$, which is not in the picture. It doesn't change anything, does it? Feb 3, 2019 at 20:06
• Actually...yeah, the third notation hasn't been talked about in the picture. It's just mentioned. Hence I asked it on here. Thanks for the explanation, I guess it is radial distance and will show up again in other chapters
– 4d_
Feb 4, 2019 at 3:05