Update on EPQ research

           Primarily, I have dedicated to this blog to my main passion of communicating ideas in mathematics to readers, and also venturing into the technical details. In this blog post, I aim to do something slightly different: I would like to explain what my actual EPQ is, and what stage I currently happen to be in.

           My EPQ is based on Maxwell's equations, and I have been doing some background research for the past month. I believe I underestimated the depth that I end up going into the mathematics behind it, and currently haven't explored the physics application of all of this research. Furthermore, I have found myself working behind schedule, even though I have been keeping at this consistently every day. I have yet to refine my topic idea in Maxwell's equations; it is still a work in progress, despite it being into mid August.
          I aim to keep it simple, and discuss in my ultimate dissertation what the importance of maxwell's equations were, and why it is a corner stone in physics. It is the first successful attempt at unifying electricity and magnetism, and the fundamental postulate in physics, that there are quantities which remain invariant under what are called Lorentz Transformations was first implied by this. Namely, the speed of light as a universal constant, which is the heart and veins of relativity, fell straight out of these equations.
        I believe I owe a quick explanation to the reader what a Lorentz Transformation is. Simply put, it is the mathematical process of switching between two non-accelerating, called inertial, reference frames, which move with a relative constant velocity to each other. As an example, if I compared what flow of time and what measurements of distance an observer moving in a space ship at half the speed of light (with no acceleration) as opposed to an observer on earth would experience, I would have to perform a Lorentz Transformation. At either case, the guiding principal behind the mathematics is the fact that some quantities are the same universally, i.e. do not vary, or in other words, remain invariant, such as the speed of light and the path between spacetime, under these Lorentz transformations. I shall end this here, lest I digress into relativity.
         
            My plan from this point on is to pick up more speed in my research. I have a five day vacation planned soon, and then the year thirteen begins with all the UCAS application process, so I really should aim to be done completely in terms of background research by then.

         In terms of my background research, I have used the following resources: Khan Academy, Yale lectures by Dr. Ravi Shanker, My Young Freedman University Physics textbook and finally a Vector Calculus book by Springer Undergraduate Mathematics Series, written by R.C Matthews.

        Let me delve into the concepts which I covered and feel fairly confident in: Understanding multi variable functions, differentiation in multivariable calculus (the partial derivative, the gradient, the directional derivative, curl, divergence, the vector valued function and the formal definition, multivariable chain rule), topics borrowed from linear algebra (span, vectors, cross and dot product, proof of these formulae), integration in multivariable calculus (double, triple integral, surface integrals, line integrals, conservative vector fields, intuition, derivation and intuition of the fundamental theorem of line integrals (which is a cousin to the fundamental theorem(s) of calculus), and flux in 2D) In physics, I have studied mainly from the Yale lectures and the uni physics textbook on the following topics, electrostatics, electric fields, symmetry arguments, and Gauss' law, which is the first of maxwell's equations. Got plenty of more content to cover, however.