# Integral notation in physics and mathematics

1. Jan 22, 2010

### 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?

2. Jan 22, 2010

### Char. Limit

Example?

3. Jan 22, 2010

### 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?

4. Jan 23, 2010

### 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 :)

5. Jan 23, 2010

### 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!

6. Jan 23, 2010

### Phyisab****

In R^3,

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

Just a matter of notation, they are equivelent.

7. Jan 23, 2010

### LucasGB

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

8. Jan 23, 2010

### 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.

9. Jan 23, 2010

### tiny-tim

Hi LucasGB!

(have an integral: ∫ )

You can't have more ∫s than ds …

so you cant 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.

10. Jan 23, 2010

### LucasGB

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?

11. Jan 23, 2010

### Char. Limit

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

12. Jan 23, 2010

### LucasGB

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

13. Jan 23, 2010

### Char. Limit

Sorry, my brain is limited to 3-D.

14. Jan 23, 2010

### 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.