Differential Forms Part 2: Differential Operators and Stokes Theorem

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 \textup{d} . I will stick to 3 dimensions for now and explain the extension into higher dimensions at the end.

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Orders of ∞

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 Part 1: Dimensions and Notation

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 »

Some Resummation Theory

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 »

Lebesgue Integration

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 »

Perturbation Theory

There exists a certain class of “hard” problems that can’t be solved with exact form. Examples include solutions to certain differential equation or higher order polynomials like quintics which can’t be solved with a simple cubic formula or quadratic formula. Perturbation theory is a tool commonly used in mathematical physics and can easily provide solutions to seemingly impossible problems. Read More »

Complex Exponentials

Complex exponentials are used immensely in math and as a result, in many fields of science. It is also used in abundance throughout this site so it is important to understand what they are for future reference. They show the relationship between exponentials and trigonometry on a fundamental level. The following is the relationship.

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Fractional Calculus

Calculus is the manipulation of one basic operator: the derivative or . This operator operates on functions and by repeatedly applying it, you can get higher order derivatives. It’s inverse operator is known as the integral. Similar to matrix operators which have eigenvalues and eigenvectors, this operator also has eigenvalues and eigenfunctions. The eigenfunction is the function which only goes through some scalar change when acted on by the operator. This scalar that the function is scaled by is called the eigenvalue of the eigenfunction. For the derivative operator, Read More »