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Season 2017
After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math, so to speak.
After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math, so to speak.
What fractal dimension is, and how this is the core concept defining what fractals themselves are.
What fractal dimension is, and how this is the core concept defining what fractals themselves are.
2017x3
Who (else) cares about topology? Stolen necklaces and Borsuk-Ulam
Episode overview
The Borsuk-Ulam theorem from topology solving a counting puzzle. Truly unexpected, truly beautiful.
The Borsuk-Ulam theorem from topology solving a counting puzzle. Truly unexpected, truly beautiful.
How e to the pi i can be made more intuitive with some perspectives from group theory, and why exactly e^(pi i) = -1.
How e to the pi i can be made more intuitive with some perspectives from group theory, and why exactly e^(pi i) = -1.
I want you to feel that you could have invented calculus for yourself, and in this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
I want you to feel that you could have invented calculus for yourself, and in this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.
2017x5
The paradox of the derivative | Essence of calculus, chapter 2
Episode overview
Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?
Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?
2017x6
Derivative formulas through geometry | Essence of calculus, chapter 3
Episode overview
A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.
A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition.
2017x7
Visualizing the chain rule and product rule | Essence of calculus, chapter 4
Episode overview
A visual explanation of what the chain rule and product rule are, and why they are true.
A visual explanation of what the chain rule and product rule are, and why they are true.
2017x8
What's so special about Euler's number e? | Essence of calculus, chapter 5
Episode overview
A visual explanation of what the chain rule and product rule are, and why they are true.
A visual explanation of what the chain rule and product rule are, and why they are true.
2017x9
Implicit differentiation, what's going on here? | Essence of calculus, chapter 6
Episode overview
Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).
Implicit differentiation can feel weird, but what's going on makes much more sense once you view each side of the equation as a two-variable function, f(x, y).
2017x10
Limits, L'Hopital's rule, and epsilon delta definitions | Essence of calculus, chapter 7
Episode overview
Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
Formal derivatives, the epsilon-delta definition, and why L'Hôpital's rule works.
2017x11
Integration and the fundamental theorem of calculus | Essence of calculus, chapter 8
Episode overview
What is an integral? How do you think about it?
What is an integral? How do you think about it?
2017x12
What does area have to do with slope? | Essence of calculus, chapter 9
Episode overview
Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.
Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the one shown in the last video.
2017x13
Higher order derivatives | Essence of calculus, chapter 10
Episode overview
A very quick primer on the second derivative, third derivative, etc.
A very quick primer on the second derivative, third derivative, etc.
Taylor polynomials are an incredibly powerful for approximations, and Taylor series can give new ways to express functions.
Taylor polynomials are an incredibly powerful for approximations, and Taylor series can give new ways to express functions.
A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
A story of pi, prime numbers, and complex numbers, and how number theory braids them together.
Can we describe all right triangles with whole number side lengths using a nice pattern?
Can we describe all right triangles with whole number side lengths using a nice pattern?
The Bitcoin protocol and blockchains explained from the viewpoint of stumbling into inventing your own cryptocurrency.
The Bitcoin protocol and blockchains explained from the viewpoint of stumbling into inventing your own cryptocurrency.
Supplement to the cryptocurrency video: How hard is it to find a 256-bit hash just by guessing and checking? What kind of computer would that take?
Supplement to the cryptocurrency video: How hard is it to find a 256-bit hash just by guessing and checking? What kind of computer would that take?
Space filling curves, turning visual information into audio information, and the connection between infinite and finite math.
Space filling curves, turning visual information into audio information, and the connection between infinite and finite math.
How do you think about a sphere in four dimensions? What about ten dimensions?
How do you think about a sphere in four dimensions? What about ten dimensions?
This episode has no summary.
This episode has no summary.
2017x22
But what is a Neural Network? | Deep learning, chapter 1
Episode overview
This episode has no summary.
This episode has no summary.
2017x23
Gradient descent, how neural networks learn | Deep learning, chapter 2
Episode overview
This episode has no summary.
This episode has no summary.
2017x24
What is backpropagation really doing? | Deep learning, chapter 3
Episode overview
What's actually happening to a neural network as it learns?
What's actually happening to a neural network as it learns?
This one is a bit more symbol heavy, and that's actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of
.. show full overview
This one is a bit more symbol heavy, and that's actually the point. The goal here is to represent in somewhat more formal terms the intuition for how backpropagation works in part 3 of the series, hopefully providing some connection between that video and other texts/code that you come across later.
A difficult geometry puzzle with an elegant solution.
A difficult geometry puzzle with an elegant solution.
Original Title: Rediscovering Euler's formula with a mug (not that Euler's formula)
A mug with some unexpectedly interesting math.
Original Title: Rediscovering Euler's formula with a mug (not that Euler's formula)
A mug with some unexpectedly interesting math.
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