Welcome back. This week, I am excited to delve into the Lebesgue integral, which is a more powerful alternative to the Riemann integral that we have dealt with so far. This new, more modern piece of mathematics is due to the work of Henri Lebesgue, a French mathematician who lived from 1875 to 1941. To actually define the Lebesgue integral, we will first develop some core ideas … [Read more...] about The Lebesgue Integral: A Newer and More Flexible Alternative to the Riemann Integral

# An Introduction to Flight

## Partial Differential Equations Meet Electricity & Magnetism: Maxwell’s Equations, Poisson’s Equation, and Eigenfunctions of the Laplacian

Welcome back. Over the last few weeks, we have studied what I hope are some fascinating results in mathematics. We have developed the theory behind multivariable calculus and applied it to describe well-behaved fluids. We have also put the classic Riemann integral on rigorous footing and proven the second Fundamental Theorem of Calculus. This week, we will continue our journey … [Read more...] about Partial Differential Equations Meet Electricity & Magnetism: Maxwell’s Equations, Poisson’s Equation, and Eigenfunctions of the Laplacian

## Predicting the Path a Particle Will Take in a Fluid: A Brief Overview

I hope that all of my readers have had a restful Thanksgiving Break and are ready to begin their final exam studies. This week, we will discuss how to design a numerical program that will predict the path a particle will take in a real-world fluid flow. To do this, we must first outline our overall problem. Suppose that we have mounted a wing inside a wind tunnel. When we … [Read more...] about Predicting the Path a Particle Will Take in a Fluid: A Brief Overview

## A Guide Through the Proof of the (Second) Fundamental Theorem of Calculus

Welcome back. This week, I hope to build on our discussion of the Riemann integral and prove the Fundamental Theorem of Calculus (FTC), which is one of the most powerful results in mathematics. This theorem technically comes in two parts and deals with antiderivatives. Essentially, an antiderivative of a function f(x) is another function whose derivative is equal to f(x). Since … [Read more...] about A Guide Through the Proof of the (Second) Fundamental Theorem of Calculus

## Constructing the Riemann Integral: A Brief Prelude to Real Analysis

Welcome back. I hope that everybody is doing well as midterm season winds down and finals season ramps up. In engineering, mathematics, and computer science, the integral is a powerful tool. A single-variable integral can yield the area under a curve, and a double-variable integral can provide an efficient way to compute the volume under a surface. Integrals also allow us to … [Read more...] about Constructing the Riemann Integral: A Brief Prelude to Real Analysis