Category Archives: Mathematics

Physics of Differentiable Manifolds: Part I

Often when trying to study general relativity, the most difficult aspect to understand rigorously is the underlying mathematics. Before anyone can do general relativity, the concepts of differentiable manifolds, smooth curves, vectors & co-vectors, tensors, tangent spaces and cotangent spaces, and many more have to be fully fleshed out. There are excellent texts out there that help achieve this goal, such as Sean Carroll’s Spacetime and Geometry. Here, however, we’ll provide a list of definitions with a few nuggets of information towards the end as a useful summary of the topics discussed in this post. Eventually, we will motivate how quantum field theory and general relativity can both be generated with concepts starting with a differentiable manifold with a metric.

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Proof of Euler’s Formula

Around 1740, Leonard Euler discovered a formula that connected functions of complex arguments to trigonometric functions, effectively forming a link between analytic functions and geometric functions which eventually extended to topology, differential equations, and mathematical physics. All of this began with one simple formula, lauded by Richard Feynman as “the most remarkable formula in mathematics,” and it is

$$e^{ix}=\cos(x)+i\sin(x), \text{ with } x\in\mathbb{R}. $$

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