[Intro Music] When you first learned about the pythagorean theorem that the sum of the squares of the two shorter sides on a right triangle always equals the square of its hypotenuse I’m guessing that you came to be pretty familiar with a few examples like the 3-4-5 triangle Or the 5-12-13 triangle, and I…

# Tag: 3brown1blue

## Cross products in the light of linear transformations | Essence of linear algebra chapter 11

Hey folks! Where we left off, I was talking about how to compute a three-dimensional cross product between two vectors, v x w. It’s this funny thing where you write a matrix, whose second column has the coordinates of v, whose third column has the coordinates of w, but the entries of that first column,…

## What they won’t teach you in calculus

3Blue1Brown [Classical music] Picture yourself as an early calculus student about to begin your first course: The months ahead of you hold within them a lot of hard work Some neat examples, some not so neat examples, beautiful connections to physics, not so beautiful piles of formulas to memorise, plenty of moments of getting stuck…

## Nonsquare matrices as transformations between dimensions | Essence of linear algebra, chapter 8

Hey, everyone! I’ve got another quick footnote for you between chapters today. When I talked about linear transformation so far, I’ve only really talked about transformations from 2-D vectors to other 2-D vectors, represented with 2-by-2 matrices; or from 3-D vectors to other 3-D vectors, represented with 3-by-3 matrices. But several commenters have asked about…

## Quaternions and 3d rotation, explained interactively

In a moment, I’ll point you to a separate website hosting a short sequence of what we’re calling “explorable videos”. It was done in collaboration with Ben Eater, who runs an excellent channel about Computer Engineering which viewers of this channel would definitely enjoy, and all the web development that made these explorable videos possible…

## What’s so special about Euler’s number e? | Essence of calculus, chapter 5

I’ve introduced a few derivative formulas but a really important one that Ieft out was exponentials. So here, I want to talk about the derivatives of functions like Two to the x, seven to the x, and also to show why e to the x is arguably the most important of the exponentials. First of…

## But how does bitcoin actually work?

What does it mean to have a bitcoin? Many people have now heard of bitcoin, that’s it’s a fully digital currency, with no government to issue it and no banks needed to manage accounts and verify transactions. That no one actually knows who invented it. Yet many people don’t know the answer to this question,…

## Why is pi here? And why is it squared? A geometric answer to the Basel problem

I’m gonna guess that you have never had the experience of your heart rate increasing in excitement while you are imagining an infinitely large lake with lighthouses around it Well if you feel anything like I do about math, that is gonna change by the end of this video Take 1 plus 1/4 plus 1/9…

## Solving the heat equation | DE3

We last left off studying the heat equation in the one-dimensional case of a rod the question is how the temperature distribution along such a rod will tend to change over time and This gave us a nice first example for a partial differential equation It told us that the rate at which the temperature…

## But what is a Fourier series? From heat flow to circle drawings | DE4

Here, we look at the math behind an animation like this, what’s known as a “complex Fourier series”. Each little vector is rotating at some constant integer frequency, and when you add them all together, tip to tail, they draw out some shape over time. By tweaking the initial size and angle of each vector,…