Before the philosopher Bertrand Russell began really thinking about the nature of sets, sets were defined informally. Set theory before various axiomatic definitions is now referred to as naïve set theory. Russell’s paradox, discovered by Bertrand Russell in 1901 was also discovered a year earlier by Ernst Zermelo but never published. This paradox shows that […]

### Proof of the determinant of a product of square matrices using Leibniz’s equation

Let us consider the square, dimensional matix. It is possible to re-express the matrix in terms of a vector containing its rows. Where It is then possible to express the product of the matrix , with another dimensional square matrix as the following, by the definition of matrix multiplication. Each row of the matrix can […]

### Can a wave be represented by infinite discrete oscillators?

In this article I will find the equation of motion for an arbitrary number of discrete coupled oscillators (I will call “beads”) on a string. Then we will take the limit as the number of oscillators tends towards infinity, and see if the wave equation can be reproduced. If so it would be a good […]

### Does Neutron Star Collapse Violate Quantum Theory?

With the death of a star meaning that radiation pressure can no longer support the gravitational crush due to its large mass, Pauli’s exclusion principle maintains an equilibrium in what has now become a neutron star. Heisenberg uncertainty allows for a neutron star to undergo ultimate collapse, as a star becomes more massive momentum space […]

### Plasma Oscillations – “Langmuir Wave” approximation derivation

In a previous article we established an approximate estimate for the frequency of oscillations in a “cold plasma” because of small disturbances in the arrangement of the electrons and ions. In this article we will establish a much better model for oscillations in a plasma, which takes into consideration the fact that Plasmas tend to be […]

### Calculus Proof of Centripetal Acceleration Magnitude

In basic circular motion in physics we are given the following equation for the centripetal acceleration on a body moving in circular motion: However how can this be proved? Well, acceleration is a vector- so what we need to know is it’s magnitude and direction. The second part of which is obvious, it’s in the […]

### Proving the Volume of a Sphere (Cylindrical Co-ordinates feat. triple integrals)

We are always taught that the volume of a sphere is 4/3πr^3 at school. But how can this be proved? One of the ways is to use cylindrical co-ordinates and integrate over suitable ranges in each of this co-ordinate system’s dimensions. But before we can do this, what are cylindrical co-ordinates? Cylindrical co-ordinates is similar […]

### Cold Plasma Oscillations – Frequency Derivation

At low temperatures it is not a bad model to take a plasma as a collection of ions and electrons, with the ions always stationary, and the electrons oscillating, due to the force upon them due to the opposing charges of the electrons and ions. For this derivation we will assume that the ions ans […]

### Schwarzschild radius – Classical Derivation

If you want to derive the Schwarzschild radius for a given body, without mastering linear algebra and tensor mathematics – don’t fear. There is a classical method, that gets to the right answer. No need to delve into the Schwarzschild metric, when you have trusty old Newton. **Disclaimer- If you can always do this derivation the […]

### The Exponential Function – Why so useful?

An exponential function is a function where the input variable (usually written as x) is an exponent. Functions with constants in the exponent are also considered exponential functions. Then what is so amazing about these exponential functions? It all comes down to the rate of change of such a function (also known as the derivative […]