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 […]

### An Introduction to Calculus of Variations

What is Calculus of Variations? In general, developments in mathematics are motivated by the need for them in applications. Calculus of variations is no exception. In fact, it was first developed in 1969 when Johann Bernoulli asked the greatest mathematical minds of his time to solve the famous ‘brachistochrone problem’. This problem involves a bead […]

### 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 […]

### 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 […]

### Linear Algebra: Derivation of formula for the inverse of a matrix and Cramer’s rule

In this article I shall attempt to derive a formula for the inverse of a matrix, and from there derive Cramer’s rule. I feel that many textbooks and courses on linear algebra (especially at high-school level) present matrices and their corresponding formulas and definitions without giving any hint of where these ideas came from, and […]

### 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 […]

### 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 […]

### Simple Time Dilation Derivation | Special Relativity

As I was interested I decided to derive the time dilation equation. My video if you’re interested in doing so yourself is linked here. To begin with I took 2 observers, one stationary and one moving. Both use a photon clock to measure the passage of time. It’s just two parallel plates, when a photon […]