#mathematical proof

Assuming product rule for differentiation is true (here’s a link to a nice little proof for that), we can get the rule for integration by parts!

If f(x) = g(x)h(x), then f’(x) =  g’(x)h(x) + g(x)h’(x)  [This is just the product rule] 

f(x) +c =  ∫[g’(x)h(x) + g(x)h’(x)].dx  [I’ve integrated both sides with respect to x indefinitely.]

f(x) +c =  ∫g’(x)h(x).dx + ∫g(x)h’(x).dx [I’ve split my integral into a sum of two integrals. The area under two curves added together is the same as adding the areas under each curve. Also, this is how you would integrate a polynomial.]

g(x)h(x) + c = ∫g’(x)h(x).dx + ∫g(x)h’(x).dx  [I’ve substituted f(x) = (g(x)h(x)]

g(x)h(x) + c - ∫g’(x)h(x).dx  =  ∫g(x)h’(x).dx [subtract  ∫g’(x)h(x).dx from both sides]

∫g(x)h’(x).dx = g(x)h(x) + c - ∫g’(x)h(x).dx [Flip it so that the equation reads in the direction we want it to. It’s ot needed; it just makes things look a little bit nicer]

∫g(x)h’(x).dx = g(x)h(x) - ∫g’(x)h(x).dx [We don’t need the “+c” because there is an integral on the right side of the equation which will add a constant anyway.]

“Pure math,” he replied. “How is that different from”—she laughed—“regular math?" Gillian asked.

“Well, regular math, or applied math, is what I suppose you could call practical math,” he said. “It's used to solve problems, to provide solutions, whether it's in the realm of economics, or engineering, or accounting, or what have you. But pure math doesn't exist to provide immediate, or necessarily obvious, practical applications. It's purely an expression of form, if you will—the only thing it proves is the almost infinite elasticity of mathematics itself, within the accepted set of assumptions by which we define it, of course.”

“Do you mean imaginary geometries, stuff like that?” Laurence asked.

"It can be, sure. But it's not just that. Often, it's merely proof of—of the impossible yet consistent internal logic of math itself. There's all kinds of specialties within pure math: geometric pure math, like you said, but also algebraic math, algorithmic math, cryptography, information theory, and pure logic, which is what I study."

“Which is what?” Laurence asked.

He thought. “Mathematical logic, or pure logic, is essentially a conversation between truths and falsehoods. So for example, I might say to you ‘All positive numbers are real. Two is a positive number. Therefore, two must be real.’ But this isn't actually true, right? It's a derivation, a supposition of truth. I haven't actually proven that two is a real number, but it must logically be true. So you'd write a proof to, in essence, prove that the logic of those two statements is in fact real, and infinitely applicable.” He stopped. “Does that make sense?”

“Video, ergo est,” said Laurence, suddenly. I see it, therefore it is. He smiled. “And that's exactly what applied math is. But pure math is more”—he thought again—“Imaginor, ergo est.”

[...]

“[The] law isn't so unlike pure math, really—I mean, it too in theory can offer an answer to every question, can't it? Laws of anything are meant to be pressed against, and stretched, and if they can't provide solutions to every matter they claim to cover, then they aren't really laws at all, are they?" He stopped to consider what he'd just said. “I suppose the difference is that in law, there are many paths to many answers, and in math, there are many paths to a single answer. And also, I guess, that law isn't actually about the truth: it's about governance. But math doesn't have to be convenient, or practical, or managerial—it only has to be true.

“But I suppose the other way in which they're alike is that in mathematics, as well as in law, what matters more—or, more accurately, what's more memorable—is not that the case, or proof, is won or solved, but the beauty, the economy, with which it's done."

“What do you mean?” asked Harold.

“Well,” he said, “in law, we talk about a beautiful summation, or a beautiful judgment: and what we mean by that, of course, is the loveliness of not only its logic but its expression. And similarly, in math, when we talk about a beautiful proof, what we're recognizing is the simplicity of the proof, its ... elementalness, I suppose: its inevitability."

“What about something like Fermat's last theorem?" asked Julia.

“That's a perfect example of a non-beautiful proof. Because while it was important that it was solved, it was, for a lot of people—like my adviser—a disappointment. The proof went on for hundreds of pages, and drew from so many disparate fields of mathematics, and was so—tortured, jigsawed, really, in its execution, that there are still many people at work trying to prove it in more elegant terms, even though it’s already been proven. A beautiful proof is succinct, like a beautiful ruling. It combines just a handful of different concepts, albeit from across the mathematical universe, and in a relatively brief series of steps, leads to a grand and new generalized truth in mathematics: that is, a wholly provable, unshakable absolute in a constructed world with very few unshakable absolutes.” He stopped to take a breath, aware, suddenly, that he had been talking and talking, and that the others were silent, watching him. He could feel himself flushing, could feel the old hatred fill him like dirtied water once more. “I'm sorry," he apologized. "I'm sorry. I didn't mean to ramble on.”

“Are you joking?” said Laurence. “Jude, I think that was the first truly revelatory conversation I’ve had in Harold’s house in probably the last decade or more: thank you.”

A Little Life by Hanya Yanagihara

The Sum of a Geometric Series Derivation

I derived this in a previous video, but I felt it was worth its own video because the formula is so important.

Video https://youtu.be/kKJFTlmuUnQ