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. Continue reading

## 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. Continue reading

## General Relativity

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. Continue reading

## Jumping in the Earth

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. Continue reading

## 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. Continue reading

## 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. Continue reading

## Heisenberg Uncertainty Derivation

The Heisenberg uncertainty principle seems like a principle that is so fundamentally experimental but it can actually be derived through theory. This requires the consideration of three concepts: the Cauchy-Shwarz inequality, measurement operators, and commutators. Continue reading

## Quantum Teleportation

Quantum teleportation is the idea that entangling states can cause very fast information transfer. The name however is misleading as this information transfer is not instantaneous but simply travels at the speed of light. The basis of this idea is based on a very interesting trick that arises out of some of the math of quantum computing. Continue reading

## Speed of Light Derivation

The speed of light may seem like an arbitrary constant of nature but, in some sense, it is actually set by other properties of the world. These other properties are the strengths of the electric and magnetic fields which are defined by the constants that are used in determination of them, otherwise known as the permittivity constant () and permeability constant (). Because light is simply an electromagnetic wave, one can derive its speed using these constants. Continue reading

## Complex Impedance

There is a certain luxury of circuit calculations for systems contain direct current that alternating current systems really do not have. It is the idea that voltage and current are “synced.” An increase in voltage will create a corresponding increase in current seemingly instantaneously. However, an alternating current that experiences voltage oscillations experiences a delay. This can mean voltage is at the highest point in its fluctuations while current only reaches such a point a little bit later at which point voltage might already be at its lowest. The ratio of voltage to current is also unclear in these circuits. This makes it hard to describe the system easily. Continue reading