Zorgoth

The previous answer provides a good basic explanation of how probability works over infinite sets. Indeed, the purpose of measure theory is primarily to properly understand integrals and probability. In basic measure theory, the relations between infinity and other numbers are specifically defined. infinity*0=0. So an integral over a set of measure zero is always zero, even if you are integrating a function which is infinite on that set, and an integral of a zero function is always zero, even if you integrate it over a set of measure infinity. This definition ONLY applies to measure theory (and specifically to the definition of the integral)! You cannot use this identity in any other context (for example, you cannot say that lim_{n->infinity}(n/n)=0 just because lim{n->infinity}(n)=infinity and lim{n->infinity}(1/n)=0). Note that measure theory kind of gets rid of ratios in probability where they don't make sense. Everything is expressed in terms of products, sums, and limits. So you don't really have to worry about dividing by infinity. In general, it's probably always a safe bet to say that 1/infinity is zero though. infinity/infinity or 0/0 is another matter. 0/0 can appear when conditioning on an event of probability zero; the way this is handled in infinite sets is with conditional probability density functions.

This is an example of the method of separation of variables for ODEs, which is perfectly standard mathematics and is taught in any college level Differential Equations course. Like the chain rule, it intuitively splits up the dx, dy, or what have you in a derivative like it was a ratio. As for what dx, dy, etc are, the precise definition comes from measure theory and/or differential geometry. For the non-mathematician, take things like separation of variables on faith and think of dx, dy, etc as an infinitesimal over which we sum when we integrate, like the delta(x) in the Riemann integral when the step size is tending to zero.

I am a PhD student and am wondering if there are good publications not relating my direct field of study that are targeted at scientists who aren't experts in a specific field. That is to say, I am looking for leisure-time science reading material that is targeted at people who know lots of math and understand science rather than at the general public. I'm not really too fussy about what sort of science or math I'm reading about. Any suggestions?

Don't waste your time on integral(e^-(x^2)) It's indefinite integral is called erf (the error function) and doesn't have a closed form (i.e. it cannot be expressed algebraically using standard functions). To prove it converges, just use a test rather than evaluating it. You *can* integrate it from -infinity to infinity using a trick from multivariable calculus, and the answer is sqrt(pi).Don't say that in your homework though, because you presumably can't prove it using calculus 2 methods.