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Unformatted text preview: Densities and Distributions Back in section 12.1 I presented a careful and full definition of the word “function.” This is useful even though you should already have a pretty good idea of what the word means. If you haven’t read that section, now would be a good time. The reason to review it is that this definition doesn’t handle all the cases that naturally occur. This will lead to the idea of a “generalized function.” There are (at least) two approaches to this subject. One that relates it to the ideas of functionals as you saw them in the calculus of variations, and one that is more intuitive and is good enough for most purposes. The latter appears in section 17.5 , and if you want to jump there first, I can’t stop you. 17.1 Density What is density? If the answer is “mass per unit volume” then what does that mean? It clearly doesn’t mean what it says, because you aren’t required* to use a cubic meter. It’s a derivative. Pick a volume Δ V and find the mass in that volume to be Δ m . The average volume-mass-density in that volume is Δ m/ Δ V . If the volume is the room that you’re sitting in, the mass includes you and the air and everything else in the room. Just as in defining the concept of velocity (instantaneous velocity), you have to take a limit. Here the limit is lim Δ V → Δ m Δ V = dm dV (17 . 1) Even this isn’t quite right, because the volume could as easily shrink to zero by approaching a line, and that’s not what you want. It has to shrink to a point, but the standard notation doesn’t let me say that without introducing more symbols than I want. Of course there are other densities. If you want to talk about paper or sheet metal you may find area-mass-density to be more useful, replacing the volume Δ V by an area Δ A . Maybe even linear mass density if you are describing wires, so that the denominator is Δ ‘ . And why is the numerator a mass? Maybe you are describing volume-charge-density or even population density (people per area). This last would appear in mathematical notation as dN/dA . This last example manifests a subtlety in all of these definitions. In the real world, you can’t take the limit as Δ A → . When you count the number of people in an area you can’t very well let the area shrink to zero. When you describe mass, remember that the world is made of atoms. If you let the volume shrink too much you’ll either be between or inside the atoms. Maybe you will hit a nucleus; maybe not. This sort of problem means that you have to stop short of the mathematical limit and let the volume shrink to some size that still contains many atoms, but that is small enough so the quotient Δ m/ Δ V isn’t significantly affected by further changing Δ V . Fortunately, this fine point seldom gets in the way, and if it does, you’ll know it fast. I’ll ignore it. If you’re bothered by it remember that you are accustomed to doing the same thing when you approximate a sum by an integral. The world isyou are accustomed to doing the same thing when you approximate a sum by an integral....
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This note was uploaded on 01/30/2012 for the course PHYS 315 taught by Professor Nearing during the Fall '08 term at University of Miami.
- Fall '08