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

## The Fourier Transform and the Heat Equation

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

## Gradients and Potential Flow Part 2: Streamlines, Harmonics, and Analytic Functions

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

## Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids

I would like to warmly welcome all of my readers back to this week’s session. Last time, we briefly discussed the relationship between Laplace’s equation (∆u = ∂2u/∂x2 + ∂2u/∂y2 = 0) and fluid flows that are irrotational and have no sources or sinks. This week, I would like to develop this topic further and build up more of the background knowledge necessary to understand it. … [Read more...] about Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids