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Q&A

$\int dx dy dz d p_x dp_y dp_z$ Does it have any physical meaning?

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I was reading a Physics book. Then I saw an equation which was looking like this :

$$\int dx dy dz d p_x dp_y dp_z$$

I was thinking it from just a Calculus book. I can see lots of variable (momentum) changing (adding all the "pieces") respect to their position. When I saw the equation a question came to my mind which is : Does it have any physical meaning if I just think it from Mathematics? Is there possible way to evaluate it?

Usually the book wrote that

$$\int dx dy dz d p_x dp_y dp_z=V4\pi (\frac{hv}{c})^2 h\frac{dv}{c}$$

I am not sure if I wrote it correctly cause I had taken the image of that book illegally that's why I couldn't take whole equation pic but I didn't notice it that time.

I didn't see this kind integral in my calculus book.

I wonder which I was reading that wrote single integral. But it I was directly searching through online I found that another had wrote 6 integral

$$\int\int\int\int\int\int dx, dy, dz, d p_x, dp_y, dp_z$$

which proves that they are taking integral for each function.

Note : I am honestly saying I don't understand anything of it now. And I may not understand answer properly also. But I am just leaving the question to read in future.. :)

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1 answer

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It's hard to answer your question specifically without the context, and obviously the physical significance of some expression depends on what the variables and operations in that expression stand for. Before considering this particular integral, I want to talk about integration generally and its notation.

When we integrate some function, we integrate it over a domain, usually some submanifold of some enclosing manifold, often $\mathbb R^n$. If we have some function $f : \mathcal M \to \mathbb R$ defined on some manifold $\mathcal M$, then we could write $\int_{\mathcal N} f(x)\mathrm dx$ for the integral of $f$ over the domain $\mathcal N$ where $\mathcal N$ is a submanifold of $\mathcal M$. For our purposes here, you don't need a super precise understanding of what a manifold is. The vague notion that it's some kind of reasonable shape like a plane or a torus is fine. As an example, we can talk about integrating over the unit disc at the origin in the plane and might notate that as $\int_D f(x, y)\mathrm dx\mathrm dy$. This would involve integrating $f$ over all the points $(x,y)$ such that $x^2 + y^2 \leq 1$. In particular, $\int_D \mathrm dx \mathrm dy = \pi$, the area of the unit disc.

In early calculus often the notation $\int_a^b f(x)\mathrm dx$ is used. This corresponds to integrating over the closed interval $[a,b]$, i.e. $\int_a^b f(x)\mathrm dx = \int_{[a,b]} f(x)\mathrm dx$. Multiple integrals are what they say, but we do have $\int_a^b \int_c^d f(x, y)\mathrm dx \mathrm dy = \int_{[a,b]\times[c,d]} f(x, y)\mathrm dx \mathrm dy$ where $[a,b]\times[c,d]$ is the submanifold of $\mathbb R^2$ whose first component is in $[a,b]$ and whose second component is in $[c,d]$, i.e. it is a rectangular area in $\mathbb R^2$.

Given this, your original integral is presumably an integral over some unspecified (at least not in the expression) $6$-dimensional (sub-)manifold (or potentially some dimension that's a multiple of $6$ depending on the dimensions of the variables, but I'll assume they're $1$-dimensional). We can also see that this integral will give us the (hyper-)volume of that (sub-)manifold in exactly the same way the similar integral above gave the area of the unit disc. Based on the naming of the variables, I assume this is a volume of some submanifold of the phase space probably for a simple particle. While not directly tangible, the sub-volumes of phase space and the preservation of the volume of phase space are extremely important facts in physical theories. For example, in the famous formula $S = k\ln W$ for entropy, $W$, in modern interpretations, is usually viewed as the volume of the submanifold of phase space compatible with some given values of macroscopic variables.

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