Multivariable Calculus

Area and Green's Theorem

Although it may seem counterintuitive, line integrals can be used to measure the area bounded by a closed curve in R2. This is thanks to Green's Theorem, and the choice of a convenient vector field with curl one. This is related to the inner works of a deviced called planimeter.

Machine Learning

The XOR Problem

A Single Layer Preceptron cannot solve the XOR problem. It can be solved using Multilayer Preceptrons, more commonly known as Neural Networks. Here is an animation of a Neural Network with one hidden layer of size 2 learning the XOR problem.

Other

Roots of Polynomials

Due to the Fundamental Theorem of Algebra, polynomials of degree n to have n roots (up to multiplicity). When the subleading coefficient of a polynomial is not zero, one of its roots will shoot to infinity as the leading coefficient tends to zero:

The Mandelbrot Set

The Mandelbrot set is the set of complex numbers c for which the iterated function f(x)=z^2+c does not diverge. Complex numbers that do diverge are colored according to how fast they diverge.