## 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.