An introduction to the quantum behavior of light, specifically the polarization of light. The emphasis is on how many ideas that seem "quantumly weird" are actually just wave mechanics, applicable in a lot of classical physics.

Exponentiating matrices, and the kinds of linear differential equations this solves.

An introduction to the quantum behavior of light, specifically the polarization of light. The emphasis is on how many ideas that seem "quantumly weird" are actually just wave mechanics, applicable in a lot of classical physics.

What is a differential equation, the pendulum equation, and some basic numerical methods

A quick explanation of e^(pi i) in terms of motion and differential equations

Divergence, curl, and their relation to fluid flow and electromagnetism

Ever tried to comb a hairy ball? Math says you failed!

A quick explanation of e^(pi i) in terms of motion and differential equations

Intuitions for divergence and curl, and where they come up in physics.

Divergence, curl, and their relation to fluid flow and electromagnetism

In this video Paul Andersen explains how a vector field shows the distribution of vector quantities. In AP Physics 1 student should be able to map and understand gravitational vector fields. In AP Physics 2 students should be able to...

What is a differential equation, the pendulum equation, and some basic numerical methods

On the divergence field In Gauss law The operator that works on a vector field.
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A normal explanation of the divergence operator, is saying that it measures how much is flowing in or out in a given area. In this video we...

Index theory is a method used to gain global information about a nonlinear differential equation. One powerful insight is that closed orbits (periodic solutions) must have at least one fixed point inside of the curve. In fact, for a...

These second-order nonlinear differential equations can be written in the form: dx/dt = f(x,y) dy/dt = g(x,y) Got a nonlinear differential equation? No problem, just linearize it! This method approximates the vector field as a linear...

These second-order linear differential equations can be written in the form dx/dt = ax + by dy/dt = cx + dy Depending on the values of a,b,c and d, the dynamics will be very different! They can be characterized by finding the eigenvalues...

These second-order linear differential equations can be written in the form

dx/dt = ax +
by
dy/dt =
cx + dy
Depending on the values of a,b,c and d, the dynamics will be very different! They can be...

An introduction to surface integrals.

Stokes's Theorem is kind of like Green's Theorem, whereby we can evaluate some multiple integral rather than a tricky line integral. This works for some surface integrals too. Let's see how it works!

Maxwell's equations are the simplest way to represent the relationship between the electromagnetic field and it's carrier particle, the photon. Simple does not necessarily mean easy though. Let's dig into what Maxwell's formulas are...

The second most beautiful equation and its surprising applications

Green's Theorem gave us a way to calculate a line integral around a closed curve. Similarly, we have a way to calculate a surface integral for a closed surface. That's the Divergence Theorem. This is also known as Gauss's Theorem, and...

These second-order nonlinear differential equations can be written in the form:

dx/dt = f
(x,y)
dy/d

t = g(x,y)

Got a nonlinear differential equation? No problem, just linearize it! This method...

Defining Green's theorem.