Calculus I/II

Some programs I have written for use while teaching introductory calculus courses.

Riemann Sums

Lets the user enter a function, and adjust the parameters for various left/right/midpoint Riemann sums.

Disks And Washers

The user may enter a function and visualize the disks-and-washers decomposition of its volume of revolution.

Cylindrical Shells

The user may enter a function and visualize the cylindrical shells decomposition of its volume of revolution.

Calculus III/IV

Some programs I have written for introducing parametric equations and fields.

Parametric Curves

A graphing calculator for parametric curves.

Parametric Surfaces

A graphing calculator for parametric surfaces.

Vector Fields

A graphing calculator for vector fields in 3-space.

Line Integrals

A graphing calculator for line integrals.

Differential Geometry

Some programs potentially useful in teaching Differential Geometry.

Geodesics on a Gaussian

Numerically solving the geodesic equation for the metric on the graph of $\exp(-x^2-y^2)$ from $\mathbb{R}^3$.

Geodesics on a Torus of Revolution

Solving the geodesic equation on a torus of revolution in $\mathbb{R}^3$ with the induced metric. Adjust the initial location and direction to explore the positively curved exterior, negatively curved interior, and the effect of the umbilic points.

Geodesic Circles

The geodesic ball based at a $p$ of radius $r$ is the set of all points which can be connected to $p$ by a geodesic of length at most $r$. Its boundary, the set of points connected to $p$ by a goedesic of length exactly $r$, is a geodesic circle. The Gaussian curvature of a surface can be defined by studying the length of these circles as a function of $r$, as $r$ tends to 0.

Gaussian Curvature

The Gaussian curvature of a surface in $\mathbb{R}^3$ at a point $p$ is the product of the two principal curvatures at that point. This program takes advantage of the existence of a relatively simple formula for this curvature on the graph of a 2-variable function to color the surface via its curvature (red=positive, blue=negative) in real time.

Conformal Torus

A conformal map is one that preserves angles, but may distort distances. This program gives some examples of distance-distorting conformal maps through visualizing conformal embeddings of the Euclidean square torus into $\mathbb{R}^3$. These embeddings come via composing an isometric embedding into $\mathbb{S}^3$ with stereographic projection.

Hyperbolic Geometry

Some programs from teaching Hyperbolic Geometry.

Structures on a Genus 2 Surface

A path in the 3-dimensional subspace of deformation space of a genus 2 surface with zero twist parameter, which approaches the boundary by almost pinching a curve.

Complex Analysis

Some programs from teaching complex analysis.

The Complex Exponential

A depiction of the graph of $z\mapsto \exp(z)$ in $\mathbb{C}^2$ projected to $\mathbb{R}^3$ via orthogonal projection with respect to some (time varying) axis.

Powers of $z$

A depiction of the graph of $z\mapsto z^n$ in $\mathbb{C}^2$ projected to $\mathbb{R}^3$ via orthogonal projection with respect to some (time varying) axis. The grid represents polar coordinates on the domain, the hues represent polar coordinates on the codomain.

Fourier Analysis

Some programs from teaching analysis.

The Ptolemaic System

In some sense the precursor to sinusoidal expansions, this Greek geocentric cosmological system successfully modeled non-circular motion as a sum of two circular motions.

A Square Orbit?

A configuration of many epicycles, giving a square orbit!

Sinusoidal Expansion

A static visualization of a square wave and its expansion into sinusoids

Complex Valued Fourier Series

What does a fourier series look like, as a function from R to C?

PDEs

Some programs from introductory PDEs.

The 1D Wave Equation

Solving the wave equation on a line with Fourier Series.

The Wave Equation on a Rectangle

Sliders allowing you to move about within a finite dimensional subspace of sinsusoidal soltuions.

2D Wave Equation: Numerically

Numerics for a gaussian wavepacket confined to a rectangular domain.

The Wave Equation on a Disk

Illustrating the Bessel function basis.

Quantum Mechanics

Some programs for single-particle quantum mechanics.

The Double Slit Experiment

Numerically solving the Schrodinger equation for a gaussian wavepacket moving towards a pair of slits in an infinite potential wall.

Elliptical Billiards

Numerically solving the Schrodinger for a free particle confined to an elliptical domain.

Quantum Tunneling

Solving the Schrodinger equation for a wave packet moving towards a finite potential barrier.