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should we have a go on lecture 20 I hate to interrupt all these great conversations uh I promise to stop after an hour uh um so this is lecture 20 which is where the main theme is the finite element method but I have uh continuation completion to to to do of lecture 19 and the you'll see the link right away uh when finite elements come I just indicated up here section 36 is where this calculus of variations is summarized and then finite elements you'll easily identify that section further along uh but this was where we were so at the end of lecture 19 we're looking for the best U of x to make this quantity small and we identified the key principle is move away a little from from U of X move away by some probably I I'm thinking of v ofx as a small perturbation so V ofx I'm thinking of as a small function in any direction positive negative and our quantity would become this but this is supposed if if U is the winner any V is supposed to give us something...

PROFESSOR: Now, what should happen? Somehow, this equation probably has solutions for all values of the energy, but those solutions diverge and are not normalizeable. It's the kind of thing you will find with a shooting method that you're doing with your computer. Count the solution and it suddenly diverges up or diverges down and cannot be normalized. But for some specific values, it can be normalized. So what we need is an intuition why this differential equation has normalizeable solutions, only if the energies, curly E, take specific values. That's intuition that we need. For that, we'll look at the equation a little closer, and try to understand what happens at the place where it can get in trouble, which is large X. That's where you expect it to get in trouble. So what does this equation become as u goes to plus minus infinity? Well this equation, at that stage, becomes like this. The second phi, the u squared is roughly equal to u squared phi. B...

You should have already downloaded and installed R on your computer. If you have not downloaded and installed R yet, please follow the instructions on the course website before continuing with this lecture. Once you've installed R, go ahead and start it. You should see the R console like we see here. This is where we'll type commands and perform data analysis. In this course, we'll mostly be working in the R console, but sometimes you'll also want to use what are known as "script files." We'll see an example of how to use a script file at the end of this lecture. In your R console, you'll always type commands after the arrow, or "greater than" sign. Let's start with some basic calculations. So with your cursor at the arrow, let's type 8, and then the star symbol for "times," the number 6, and hit Enter. You should see the result 48, or 8 times 6. In this way, R can be used as a basic calculator. We can perform ...