I always think of Los Angeles in the present, if that makes sense. As if the Los Angeles that I know is the only Los Angeles that ever was or will be. But, that’s not true. It’s often easy to forget that the very ground beneath our feet has been around long before me. This soil has been passed from generation to generation, changing a bit with each pocket of passed time. My … [Read more...] about Exploring Los Angeles Through XR

# Articles

## The Calculus of Variations, the Euler-Lagrange Equation, and Classical Mechanics

Welcome back. This week, we will take a short break from partial differential equations and have a brief foray into the calculus of variations, a field of mathematics that is concerned with optimizing functionals. A functional, essentially, is a real-valued function that takes functions in as inputs. For example, F[f] = ∫[0,1] f dx, is a functional that takes in a function, f, … [Read more...] about The Calculus of Variations, the Euler-Lagrange Equation, and Classical Mechanics

## Higher Dimensional Integration By Parts and Some Results on Harmonic Functions

Welcome back. Integration by parts is a very useful technique that usually shows up in introductory calculus courses. It allows us to efficiently integrate the product of two functions by transforming a difficult integral into an easier one. When working with a single variable, the integration by parts formula appears as follows: ∫[a,b] g(x) (df/dx) dx = g(b)f(b) - g(a)f(a) … [Read more...] about Higher Dimensional Integration By Parts and Some Results on Harmonic Functions

## Separation of Variables and the Method of Characteristics: Two of the Most Useful Ways to Solve Partial Differential Equations

Welcome back. I hope all of my readers are having an excellent start to the new semester. Over the last couple of months, we have discussed partial differential equations (PDEs) in some depth, which I hope has been interesting and at least somewhat enjoyable. Today, we will explore two of the most powerful and commonly used methods of solving PDEs: separation of variables and … [Read more...] about Separation of Variables and the Method of Characteristics: Two of the Most Useful Ways to Solve Partial Differential Equations

## The Lebesgue Integral: A Newer and More Flexible Alternative to the Riemann Integral

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