The Heaviside step function shows up just about everywhere, with its integral representation and Fourier transform often cropping up in quantum field theory and signal processing. These expressions often lack explanation, so the purpose of this post is to illuminate their origins. Prerequisites: To understand this post you’ll need to have a passing familiarity with Fourier transforms, distributions, and complex analysis. Recommended Reading: This excellent answer by user Patch on the math StackExchange. The naive approach As we know, the Fourier transform \(\tilde{f}(k)\) of a function \(f(x)\) is given by $$\tilde{f}(k) = \int_{-\infty}^{\infty}f(x)e^{-ikx}\mathrm{d}x$$ This problem isn’t hard—presumably, the transform of the Heaviside function is defined as $$\tilde{\theta}(k) = \int_{-\infty}^{\infty}\theta(x)e^{-ikx}\mathrm{d}x.

This semester, a controls class I’m enrolled in has promised to be proof-based. Writing proofs is not a muscle I exercise often, so I figured it would be best to get back into things with some proofs of elementary theorems in linear algebra. This is not going to be a particularly exciting post—it’s mainly done for the benefit of myself. Prerequisites: To understand this post, you should be familiar with linear algebra. Recommended Reading: Linear Algebra Done Right by Sheldon Axler. Finite Dimensional Linear Systems by Roger Brockett. Linear dependence and span Theorem: Given a vector space \(V\) over a field \(\mathbb{F}\) and a spanning set \(V_0 = \lbrace v_1, \dots, v_n\rbrace\) with \(v_i \in V\) for \(i = 1,\dots,n\), an arbitrary linearly indepdendent set of vectors \(U = \lbrace u_1, \dots, u_k\rbrace\) with \(u_j\in V\) for \(j=1,\dots,k\) has at most \(n\) elements.

If you’ve spent time with analytical mechanics, you’ve probably stumbled upon the concept of a symplectic manifold. In particular, you’ve probably heard how Hamiltonian mechanics is naturally described by a symplectic manifold. Unfortunately, many sources explaining why this is so are opaque to the uninitiated or painfully slow to get to the point. I’ll try to outline the geometric picture of mechanics using a combination of qualitative and quantitative arguments. This borrows quite a bit from Henry Cohn’s note on the topic, so I recommend you check it out. Prerequisites: To understand this post, you should be comfortable with differential geometry and its two most popular notations: indicial notation and coordinate-free notation.

During her Master’s program, my girlfriend signed up for a Putnam Competition preparation course. She doesn’t have a background in math, but she thought it would be fun—and it was! There were lots of interesting practice problems we solved together and this was my favorite. Prerequisites: To understand this post, you should have some knowledge of modular arithmetic. Recommended Reading: The Putnam Competition website. The setup Imagine a room with \(N\) prisoners. They are going to play a game for their freedom. They can talk amongst themselves and come up with a strategy before the game starts, but as soon as it begins they are immobile and unable to communicate.

In many space physics papers investigating nonadiabatic current sheet scattering, the simplified Harris model is used. In Cartesian coordinates, it takes on the form $$\mathbf{B} = B_x\tanh\Big(\frac{z}{L}\Big)\hat{\mathbf{i}} + 0\hat{\mathbf{j}} + B_z\hat{\mathbf{k}}$$ where \(L\) parameterizes the current sheet thickness. This is nice approximation for a relatively static magnetosphere, but it doesn’t capture a thinning current sheet. For that, we’ll need to make the change \(L=L(t)\). In turn, this will induce an electric field. What is the expression for that field? Prerequisites: To understand this post you’ll need to be familiar with Maxwell’s equations. To understand the motivation of this post, familiarity with basic space plasma physics and the structure of the magnetosphere is necessary.

Recently, a friend of mine shared with me the idea of the Leibniz isochronous curve. In words, this is the trajectory a frictionless object would follow if its vertical velocity was left unchanged by a constant, downward force. Using the standard \(x\)-\(y\) Cartesian axes, the curve is parameterized by $$\begin{aligned}x &= x(t)\\y &= -v_0t\end{aligned}$$ where \(v_0\) is positive. What is the exact expression for \(x(t)\)? Prerequisites: To understand this post you’ll need to be comfortable with the Lagrangian formulation of classical mechanics. Recommended Reading: The Variational Principles of Mechanics by Cornelius Lanczos. The Theoretical Minimum by George Hrabovsky and Leonard Susskind.