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Opera, that's a good one. That's good. It was opera, yes. What opera? No, it wasn't Wagner, but that's a good guess. It's certainly a reasonable style, way over the top. That was Maria Callas singing La Mamma Morta from Andrea Chenier, which was written in 1895, and premiered in the spring of 1896 exactly at the time when the world was going nuts over this mysterious form of radiation that can see inside the human body. So, I thought that was a good match. The thing's best in class. That's Maria Callas. This is opera for those of you who don't like opera. This, by the way, some of you may recognize if you saw the movie Philadelphia. This is the piece that's playing when the Tom Hanks character visits the loft, or excuse me, the Denzel Washington character visits the loft of the Tom Hanks character. And this is playing. It's a fantastic piece, way over the top. And this is Maria Callas who is the woman that restored melodrama to oper...

Okay, that's, so to speak, the text for today. The Fourier series, and the Fourier expansion for f of t, so f of t, if it looks like this should be periodic, and two pi should be a period. Sometimes people rather sloppily say periodic with period two pi, but that's a little ambiguous. So, this period could also be pi or a half pi or something like that as well. The an's and bn's are calculated according to these formulas. Now, we're going to need in just a minute a consequence of those formulas, which, it's not subtle, but because there are formulas for an and bn, it follows that once you know f of t, the an's and bn's are determined. Or, to put it another way, a function cannot have two different Fourier series. Or, to put it yet another way, if f of t, if two functions are equal, you'll see why I write it in this rather peculiar form. Then, the Fourier series for f is the same as the Fourier series for g. And, the reason is because if...

PROFESSOR: Since z, that's straight up. And that is going to provide for you a cell that has the shape of a square prism. This would be a1. This would be a2. And this would be a3-- I'm sorry, this would be c. And if you took the choice for the third translation as 1/2 half of a1 plus 1/2 of a2 plus some amount z up above the base of the cell, and then redefined a T3 prime that would be equal to-- sorry. What am I doing here? Yeah. This is right. 1/2 of a1 1/2 of a2, and z straight up. Define a T3 prime as 2T3 minus a1 minus a2. And that will define for you, again, a cell in the shape of a square prism with a1 and a2. And now the translation that went up directly over the center of the base of the net below. Twice that minus a2 minus a1 brings you back to a third translation T3 prime that's normal to the base. So both cells have the same shape. This is a primitive. This is a body-centered tetragonal lattice. And the symbol that's used to represent the body-...

missed out on one that's quite fun uh and uh it leads to uh these fantastic pictures of fractals you may have seen I intended to bring in a book by pipin that has uh sort of the beauty of fractals is his title has pictures of the mandal Bro Set uh just amazing um and just it's interesting mathematics and it's mathematics of recent years so I'm going to put off the wave equation and heat equation to the second lecture today lecture 12 and discuss this one because it's sort of it's so here's the general plan you have some function and you start at some point u0 and you compute F of u0 and that gives the next point then you compute F of U1 and that gives U2 and then you keep going so so u n is so to speak F of f of f of it's where're we're uh we're taking well the mathematical word is composition so so let me just put up U2 so you see U2 U2 is f of f of U uo because that's U1 so we're we're repeating the same operation ...