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hughbg · 2008-04-28 11:58:48

I'm referring to the equation for a distribution of charges (E) in the section Some Practical Applications in a wiki page: http://en.wikipedia.org/wiki/Multiple_i … lications.

Up until that point the page refers to multiple integrals of the form $\int{} \int{} \int{} dx dy dz$ , which I can understand. Then all of a sudden we have $\int{} ... d^{3}r$ .

What does $d^{3}r$ mean? And why aren't there three integral signs if it's a triple integral?

M@Man · 2008-04-29 18:04:47

hughbg,

Don't be confused by the (sometimes awkward) notation. Both expressions you have written stand for the same volume (triple) integral:

$\int{}_V d^3 r = \int{} \int{} \int{} dx dy dz$

There are some advantages to using either notation. The three integrals over x, y, and z are somewhat clearer, and it is easy to see how to perform those integrals. However, this method requires integrating the volume in Cartesian coordinates, which may not always be the most convenient coordinates for your integral. If you were doing a volume integral over the interior of a sphere, for instance, doing the integration in Cartesian coordinates - although it must give the same answer - would be significantly harder than doing the integral in spherical coordinates. This is an advantage of the single-integral notation: it is expressed in a form that is independent of coordinates and lets you manipulate it easily before you have to actually pick a coordinate system before you do the integral. Carrying around the three integrals, on the other hand, would just be cumbersome.

And the differential being integrated over on the left-hand side is merely the volume element:

$dV = dx dy dz = r^2 sin \theta dr d\theta d\phi = d^3 r$

The middle two expressions are in Cartesian and spherical coordinates, respectively.

So don't be dismayed by the strange notation. The mathematics it stands for is the same triple integral that you are familiar with; the notation just evolves as you need to use it in a variety of contexts. The $d^3 r$ notation, for instance, becomes useful when you need to do integrals in more than 3 dimensions. When doing special relativity, you often integrate over 4-dimensional spacetime with volume element $d^4 r = (dx) (dy) (dz) (c dt)$ (where $c$ is the speed of light). In general, the notation for the volume element is $d^D r$ , where $D$ is the number of dimensions being integrated over.

And a note on the bounds for the integrals: especially when using the $\int{}d^3r$ notation, the volume being integrated over is usually implied. Often the volume is all space, but sometimes we actually mean to imply integration over a particular volume that is understood. Similarly, we often leave out the bounds of integration on a 1-dimensional integral $\int{} dx$ when we mean to integrate from $x = -\infty$ to $x = +\infty$

Chris · 2008-04-30 03:37:38

To give you a good idea why the shorthand is useful, in statistical mechanics we will integrate over the phase space volume for a system to obtain the number of possible microstates. You have N particles ( $10^{23}$ ) that each have 3 position coordinates (configuration space) and 3 momentum coordinates (momentum space) in phase space. That's $6 \times 10^{23}$ mutually perpendicular axes:

$\Gamma = \frac{1}{h^{3N}}\int \int \int ... \int \int \int dx_1 dy_1 dz_1 dx_2 dy_2 dz_2 ... \\ ... dx_{N} dy_{N} dz_{N} dpx_1 dpy_1 dpz_1 dpx_2 dpy_2 dpz_2 ... \\ ... dpx_{N} dpy_{N} dpz_{N}$

It's much easier to write:

$\Gamma = \frac{1}{h^{3N}}\int_V \int_V dV_{c} dV_{m}$

or:

$\Gamma = \frac{1}{h^{3N}}\int_V d^{3N}r$