In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this. In fact, the drawn out results from the posts The Shape of a String and The Lagrangian are just two cases of the one equation.
In the first post, we established a general intuition of how forms work and why they may provide a better geometric intuition of what is actually occurring. It was mentioned that these ideas extend the ideas of vector calculus so it seems natural to see how differential operators like gradient, curl, and divergence arise in the context of differential forms. It all comes out of the analysis of the exterior derivative . I will stick to 3 dimensions for now and explain the extension into higher dimensions at the end.
Math is cool. Here are some cool things in math that I don’t think are really extensive enough for their own post but I still want to share.
The following is a great reason why.Read More »
Calculus of variations is an extremely useful and amazing tool in physics, math, computer science, and a variety of fields. Similar to how regular calculus is focused around functions and differentials, this field focuses on functionals and variations. A functional takes in a function and spits out a number. The following are examples of functionals.Read More »
The idea of infinity is easy to look over upon first glance. It can simply be defined as the idea that numbers go on forever and it is easy to end there. However, there are very developed and well-defined notions of infinity suggest that there are different orders and types of infinity which come with various properties. The consideration of all these are vital to our understanding of infinite quantities and especially in set theory. The exploration of these transfinite numbers Read More »
Differential forms is a topic that, in some sense, extends ideas presented in vector calculus with more suggestive notation and geometric intuition into higher dimensions. The distinction may seem small and insignificant especially in the third dimension that we live in but its results and implications are quite elegant and can lead to nice formalization of certain results such as Stokes’ Theorem. Read More »
Perturbation theory, as mentioned in an earlier post, is a very important part of the study of many fields but a recurring problem is the issue of summing divergent sequences which sometimes arise in a solution. Even some convergent solutions are very hard to sum because we can only calculate the first two or three terms in a reasonable amount of time. As a result, the study of these infinite summation become very important to advance the field.Read More »
General Relativity is perhaps one of the most enlightening theories in all of physics that reconciles many fundamental ideas about gravity, mass, and spacetime. The understanding of such a theory requires, in some sense, a certain conceptual jump which in the past has been explained through a “stretchy fabric” where mass acts as wells and other masses fall into these wells acting as a conceptual picture of general relativity. If you don’t know what I’m talking about, watch this video. This is immensely flawed in many ways. Read More »
This is just a quick fun article that may be trivial to some readers but will probably be very interesting to others. Imagine you jump through a hole that goes through the center of the earth like the little man below. Read More »
Integrals are a great point of interest in many areas of mathematics and, when learned about, are often overlooked on the fundamental level. The ideas of Riemann integration, which is what many learn about, are very vast and complex and can provide powerful results but there exists, in some sense, a better and more general form of integration that can account for scenarios Riemann integration cannot. This is Lebesgue integration.Read More »