An Introduction to Flight

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

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

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

Welcome back. In this week’s blog, we will explore the Fourier transform, which is a valuable tool across engineering disciplines. Like most tools, its primary power is to make complicated problems simpler. To demonstrate this, we will first define the Fourier transform and establish some of its most important properties. We will then apply it to solve the one-dimensional heat … [Read more...] about The Fourier Transform and the Heat Equation

Last time, we learned that if we have a scalar function φ(x,y) that satisfies Laplace’s equation (∆φ = ∇ • ∇φ = 0), then its gradient can define the velocity field of a well-behaved fluid that is irrotational and has no sources or sinks. Today we will try to find equations for the streamlines that particles have as they move through the fluid flow defined by ∇φ. Here, we can … [Read more...] about Gradients and Potential Flow Part 2: Streamlines, Harmonics, and Analytic Functions