Integral notation in physics and mathematics

[LucasGB]

Why is it that every time I find an area or volume integral in my physics books, they are written using only one integral sign, while in calculus books they are written with two or three integral signs, respectively? Which one is more correct and rigorous?

 

[Char. Limit]

Example?

 

[LucasGB]

In calculus books, vector flux is written as two integral signs (E . da). In physics books, vector flux is written as one integral sign (E . da). It's the concept of multiple integrals in calculus, which even though it is applied extensively to physics, it's done in a different notation. Which one is more correct?

 

[elibj123]

It's not a matter of what is more rigorous or correct, it's a matter of notation. I guess that in a math exam, you'll be expected to write as many integral signs as needed, and to specify which integral corresponds to which variable, and in a physics exam they just won't care.

I guess physicts don't want to waste time on writing many integral signs :)

 

[Ben Niehoff]

It depends what branch of mathematics you're talking about. Once you get into differential forms, you typically write only one integral sign, and often you are integrating over a space of arbitrary (but finite) dimension!

 

[Phyisab****]

In R^3,

$$\int_{V} dV =\int\int\int dx dy dz$$

Just a matter of notation, they are equivelent.

 

[LucasGB]

OK, so I take from this that I can write it the way I think looks best?

 

[Phyisab****]

Well the first way is more convenient and brief for some manipulations, but if you are going to explicitly carry out the triple integration, you will probably want to start by writing it like that. So yea write it whichever way seems good to you.

 

[tiny-tim]

Hi LucasGB!

(have an integral: ∫ )

You can't have more ∫s than ds …

so you can't have ∫∫∫ dV, but you can have ∫∫∫ dxdydz​

and you can't have less ∫s than ds except that, in my opinion , you can have only one ∫, purely to save space.

 

[LucasGB]

You can't have more ∫s than ds …

so you can't have ∫∫∫ dV, but you can have ∫∫∫ dxdydz​

and you can't have less ∫s than ds except that, in my opinion , you can have only one ∫, purely to save space.

That's important, I didn't know that. Well, alright then, I'll stick to the single ∫ notation, at times I don't have to explicitly carry out the integral. And another thing, the integral symbol with the small circle in it stands for a closed integral, right? So for a line integral, it means you take the integral over a closed loop, and for a surface integral, it means you take the integral over a closed surface. But what about a volume integral? What the hell is a closed volume?

 

[Char. Limit]

I would guess that a closed volume will find the integral of a hypervolume in $$\mathbb{R}^4$$.

 

[LucasGB]

Yes, but what IS a closed volume? What's the difference between a closed and an open volume?

 

[Char. Limit]

Sorry, my brain is limited to 3-D.

 

[LucasGB]

Oh, I get what you're saying. A closed line forms a surface, and a closed surface forms a volume. Therefore, a closed volume would form a hypervolume. I'm sorry, I hadn't grapsed that before, and was under the assumption that closed and open volumes were objects that could be understood in 3D. I get it now, thanks.