Recent content by theorem4.5.9

Thanks for replying. A proper text on the subject is probably the way to go, but I don't have the time to invest in that right now, but I think I can get a lot for very cheap with QED. Correct me if I'm wrong, but I think the heuristics are much more than a rule of thumb. I'm under the...

Feynman's first topic in his second lecture on QED is the nature by which light reflects off of a mirror. We work in ##\mathbb{R}^2##. Suppose we have a light source sitting at ##(-1,1)## and a photomultiplier sitting at ##(1,1)##, with a mirror along the x-axis. We also place a block between...

Perhaps you are interested in covering theorems? You may start with the 5-r covering theorem (it's one of the most basic and easier to understand), then maybe the Vitali or Besicovitch covering theorem, though they get very technical. Though your phrasing makes me think that you have some kind...

When you asked One entity that you named was a sigma algebra, i.e. ##\sigma (C)##. It's a sigma algebra because the set that ##\sigma (C)## represents satisfies the sigma algebra axioms. The second entity you named, ##C## is not a sigma algebra. It's a set with no special properties. However...

This is poorly worded. A sigma algebra is a sigma algebra, regardless of where it comes. I believe your confusion is with the generating set. ##C##. ##C## is not a sigma algebra, but there is a unique minimal sigma algebra containing ##C##, you've denoted it ##\sigma (C)##. In this sense, the...

Yes, a description of the problem would be very helpful, as the isoperimetric problem has a number of settings and generalizations. Frank Morgan's Introduction to Geometric Measure Theory is a good start, specifically chapter 5 which gives an outline of the compactness theorem. If you're...

As the general answers say, the statement is rigorously meaningless unless used in very special (and typically advanced) scenarios. On an intuitive infinitesimal level, I'd say it means that ##x## is constant as algebrat says.

No, there are no mathematicians who travel for all time. Every mathematician has a finite distance to travel, and will arrive in finite time. It seems your confusion parallels a lot of students confusion between convergence vs uniform convergence (say for example of sequences of functions)...

Contrary to how math is typically presented, most subjects are not linear in pedagogy. Your question is about preference, and that changes with different people and authors. Personally I like to use them early because they have a very geometric description to them.

Would you mind elaborating on this? I've worked out the case where ##f## is injective below, but I don't see how one would define composition where ##f## vanishes at more than one point. If ##d, g, f## are test functions and ##f## is injective and ##f(x_0)=0## then $$\langle d\circ f, g...

The obvious answer is to ask the professor teaching the course. I'm sure he'd be happy to help you.

Determinants are used all over the place, not only in linear algebra. One of the uses of determinants that comes up a lot in my studies is its use in computing areas. In calc 3 you learned (or will learn) that in order to do a u-substitution in 3 dimensions, you need to multiply dx by the...

Transfinite induction is very close in spirit to that of strong induction. You assume that a statement ##P(x)## is true for every ##x<a##, and then show it's true for ##x=a##. In typical induction, you are only concerned with the whole numbers, though in transfinite induction you instead look...

There are mathematical spaces which have components that are "infinite" distances apart. An easy example is to take two compact (closed and bounded) disconnected components in ##\mathbb{R}^2##. Define the distance between any two points to be the minimum length of all paths that live in the two...

I am not aware of such an identity. What is the sum over? The dirac delta function for a single variable is simply ##\delta [f(x)] = f(0) ##. In two variables it would be ##\delta [f(x,y)] = f(0,0)##. Of course you can shift the delta function to any point off the origin as well.