Mathematics

In my opinion, one of the reasons that differential forms are so confusing to new students is that the pedagogy in introductory calculus classes is often insufficiently rigorous. For example, in a first course in calculus, students are taught to integrate single variable functions and are instructed to think of this operation as computing the area under a curve.

The Euler-Lagrange Equation In the previous post, we introduced the variational calculus and took the first steps towards deriving necessary and sufficient conditions for establishing local extrema of functionals. In this post, we will build on our previous work and derive the Euler-Lagrange equation, then subsequently apply it to some physics problems.

In introductory calculus, we learn to take derivatives of functions of a single variable. In vector calculus, we learn to do the same for vector-valued functions, although here the analysis becomes more complex as there is no longer a single, canonical notion of a derivative.

An explanation using Sard’s Theorem.

Work In Progress! A manifold in the broadest sense, is a structure which is locally homeomorphic to $\mathbb{R}^n$ at each of its open sets. In layman’s terms, this means that while a manifold may be sufficiently “abstracted” such that its elements look nothing like those of $\mathbb{R}^n$, we can regardless treat the manifold as if it’s $\mathbb{R}^n$, with a few potential caveats.

Any continuous function can be approximated to an arbitrary degree of accuracy by some neural network.