Also check out Elementary Calculus: An Infinitesimal Approach for a mathematically rigorous course in infinitesimal calculus. I think it is much more intuitive than typical limit calculus.
If you already understand limit proofs of calculus go ahead and read the first chapter and the epilogue to get a pretty complete picture on infinitesimal calculus.
The "multiplying changing numbers" analogy made me think of "integrating really is just calculating the average", ie integrating f(x)dx is like taking the average of f(x) and multiplying by the size of the interval.
Yeah that's exactly what it is! In probability theory, the expectation of a random variable is just its integral with respect to a probability measure.
I don't get the "revelation" he's claiming that this is. When you're finding work or distance, two examples he gave, you're still finding the area under a curve. Sure, the axes will be representing time and velocity when calculating distance, or force and distance while calculating work, but that is the exact same thing as calculating area when the axes are used to represent distances of width and height.
He's right that's it's akin to multiplication. In fact, you can find the areas under trapezoids using very simple integrations, but he did nothing to discount the fact that definite integrals are all about areas under the curve.
Hi, thanks for the comment. My main point is that going beyond the "area" understanding leads to a better understanding of calculus:
* Finding area does not readily imply an inverse operation, but multiplication does ("division" is differentiation, and it is the inverse). Most students would not posit the existence of an inverse of "finding the area", but multiplication is easy to reason about.
* Multiplication scales to N variables (multiple integration), while the mental model of "area" breaks down after the 3rd dimension.
* Multiplication matches more closely to what is meant in a lot of cases. When you integrate speed and time, you are trying to multiply them. You probably don't mean to imply that you're going to plot them on a graph and take the area under the curve as distance. That may be the mechanics of it, but finding area is not the purpose of the operation.
So, area isn't wrong by any means, it's just a limiting viewpoint in my mind, and there are better analogies out there. Thanks for the comment!
Seems to be helpful for physicists, not so much for mathematicians. A physicist will look at an integral and say "oh, yeah, charge is field times area, that makes sense!", while a mathematician isn't going to say the same about measures. I think that's the "revelation" he was going for.
BTW, I always thought of convolution as multiplication. Anyone else feel cheated by this guy for that reason?
Not just you. I seems much more natural to me to think of an integral as "continuous sum."
You can often describe the exact same thing using a discrete sum or an integral, e.g., in a classical mechanics class you first consider a system of N connected springs, and a number of properties (like, say lagrangian, energy etc.) of the system will be expressed as sums with N terms. Then you take the continuous limit, the connected springs become one string and voila, all sums become integrals.
In quantum mechanics for some systems the set of all energy levels is discrete (e.g., the harmonic oscillator), for some it's continuous (e.g., a free particle), and often it is part-discrete part-continuous (a hydrogen atom.) Many formulas will require you to take a sum for the discrete part and an integral for the continuous part. Some quantum mechanics textbooks introduce a special symbol, a capital sigma overlaying an integral symbol, which means "integrate or add, whichever appropriate."
Maybe it's just me, but I found Integrals more intuitive as a continuous "sum" of probabilities in a distribution. So, if you want to marginalize a discrete distribution, you just add all its components over the conditional variable. If you want to marginalize a continuous distribution, you integrate.
I guess you could say that you're simply enumerating infinitesimal densities?
http://www.math.wisc.edu/~keisler/calc.html