Tag: physics

Toppling the Math Idols: Uncovering Parallel Transport Within the Riemann Tensor

Curved grid surface with arrows marking loop segments A, B, C, and D
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A famous 17th century defender of the newly emergent Scientific Revolution, Francis Bacon in his 1620 book Novum Organum also warned about the four main ways that we can obstruct truth. One of these obstructions is called the "Idols of the Marketplace," which is how our imprecise and deceptive use of language may blind us. … Continue reading Toppling the Math Idols: Uncovering Parallel Transport Within the Riemann Tensor

Rolling the Dice: A Derivation of Schrodinger’s Equation from the Wave Equation

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The quantum world reveals an astonishing truth: light, electrons, and indeed all matter in the universe — including ourselves — are described by mathematical wavefunctions that represent the probabilities of finding each particle in a particular location. These wavefunctions remain spread out until a measurement or interaction with the environment, which causes them to entangle with the environment, destroying observable interference and yielding a discrete location for the particle.

How the Curl of Curl Gives Light on Electromagnetic Waves and other Phenomena

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It is amazing that light is actually an electromagnetic wave, but how can we actually "illuminate" this through mathematics? It was James Clerk Maxwell back in the 19th century who built upon his predecessors' scientific observations of electric and magnetic fields and then applied mathematical analysis to these observations to enlighten us all about the … Continue reading How the Curl of Curl Gives Light on Electromagnetic Waves and other Phenomena

Divergence, Curl and the Taylor Series Approximation

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I wanted to examine two crucial principles in understanding the physics behind how systems flow- divergence and curl- and to fully explain both the mathematical assumptions and intuitions that underlie these bedrock foundations of field motion. I think what is most often overlooked in the explanations of divergence and curl is the fundamental importance in … Continue reading Divergence, Curl and the Taylor Series Approximation

Of Unwavering Importance: The Wave Equation Derivation

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(Also, note that to achieve these boundary conditions, the sine waves across L must all be increments of half wavelengths- this is covered more in the Appendix section.) (Note: in the above the imaginary number i was dropped from multiplying the Dn coefficient. This can be done by constructing the original Cn and Dn coefficients … Continue reading Of Unwavering Importance: The Wave Equation Derivation

The Lorentz Factor and the Invariance of Relativity

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In many ways, the theory of relativity could have been named the theory of constancy, since it relies upon things that are invariant like the speed of light and the Lorentz factor. This post aims to derive the mysterious Lorentz factor, Spacetime Invariant for flat space, and Spacetime Coordinate Transforms through an intuitive, geometrical approach … Continue reading The Lorentz Factor and the Invariance of Relativity

Orbiting Around the Truth Part II: Adjusting the Orbital Path Equation Using General Relativity

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In the previous post https://mathintuitions.com/2024/08/14/orbiting-around-the-truth-approximations-involved-in-newtons-and-einsteins-orbital-equations-part-i/, we were able to derive the elliptical orbital path of a planet, where its radial length r is a function of its angle, phi: where L, the angular momentum, is equal to Before arriving at the above expression for r in terms of phi, we arrived at a simpler, more … Continue reading Orbiting Around the Truth Part II: Adjusting the Orbital Path Equation Using General Relativity

Imaginary Numbers and Wormholes

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     In Paul J. Nahin's book "An Imaginary Tale," a very interesting mathematical puzzle is introduced regarding imaginary numbers that leads to connections to wormholes in the space-time continuum. (image of wormhole is from Wikipedia AllenMcC. Vector: KES47 - Own work based on: Lorentzian Wormhole.jpg by AllenMcC., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=30231987 ). He discusses the discovery of … Continue reading Imaginary Numbers and Wormholes