# Gallery — Math

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Using the One Dimensional Wave Equation to Represent Electromagnetic Waves in a Vacuum
The differential wave equation can be used to describe electromagnetic waves in a vacuum. In the one dimensional case, this takes the form $\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 0$. A general function $f(x,t) = x \pm ct$ will propagate with speed c. To represent the properties of electromagnetic waves, however, the function $\phi(x,t) = \phi _0 sin(kx-\omega t)$ must be used. This gives the Electric and Magnetic field equations to be $E (z,t) = \hat{x} E _0 sin(kz-\omega t)$ and $B (z,t) = \hat{y} B _0 sin(kz-\omega t)$. Using this solution as well as Maxwell's equations the relation $\frac{E_0}{B_0} = c$ can be derived. In addition, the average rate of energy transfer can be found to be $\bar{S} = \frac{E_0 ^2}{2 c \mu _0} \hat{z}$ using the poynting vector of the fields.
Eric Minor
Building new topological spaces through canonical maps
Suppose we have some topological spaces lying around. How can we build new topological spaces using the old ones? There are four fundamental constructions: subspaces, disjoint unions, products, and quotients. Defining the topologies on each can be done in two ways. One way is through ad hoc definitions. These definitions make some intuitive sense, but look very different from one construction to the next. The other way uses canonical maps. Canonical maps provide a single framework in which all constructions obey the same unifying principle.
Sean Raleigh
FSU-MATH2300-Project3
This is a project to develop students' understanding of Newton's Method using the tools available in Geogebra. This project was adapted from a similar project developed by folks at Grand Valley State University. (If any of you see this and would like more specific attributions, please let me know.)
Sarah Wright
Sine approximation
A 5th order sine wave approximation
Pepijn de Vos de Vos
Time-Frequency Analysis and the Wavelet Transform
A brief (lets not kid ourselves its long) introduction to the continuous and discrete wavelet transforms. Comments on implementations on the computer using MATLAB and other software is also included.
ryanplestid
When Area and Perimeter are “Equal”
Various geometrical shapes are described, for which the numerical value of the perimeter is the same as that of the area. Cases of one or two parameters are explored.
Rick Powers
Math 333 Weekly Homework Template
This is a template for students from Math 333 at Hood College to use for weekly homework submission. (Fall 2018)
Jill Tysse
What is the maximum altitude reached by a Superpressure balloon?Can we control the balloons altitude with an air pump?
A detailed report of findings on the altitudes which can be reached by super pressure balloons and how various factors and considerations affect this. Superpressure balloons are deployed and researched by various organisations including NASA, to solve technical limitations such as cell tower coverage as well as advancing fields of research. Balloons are used in planetary exploration, and weather prediction to teaching primary school physics. The versatile yet simple aerostat has been a valuable tool in many areas of engineering and their altitude ceiling is of great scientific interest. To solve the problem without the ability to physically reproduce the scenario, required mathematical models to be created as a means of simulating the effects of real world physics. A degree great enough to output an accurate and hence useful result without becoming too complex to be computable is the fine balance attempted to be created by this paper.
Charles Poppy
Math 53 Homework template
For math 53 homework
Jupiter Zhu