Tag: physics

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

Featured ~ Josh Fialkow ~ 2 Comments

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

Featured ~ Josh Fialkow ~ Leave a comment

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

Featured ~ Josh Fialkow ~ Leave a comment

(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

Featured ~ Josh Fialkow ~ Leave a comment

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 using the fewest assumptions possible along with the most mathematical proof possible. Before relativity … Continue reading The Lorentz Factor and the Invariance of Relativity

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

Featured ~ Josh Fialkow ~ Leave a comment

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

Featured ~ Josh Fialkow ~ Leave a comment

     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

Stranger than Fiction: A Derivation of the Coriolis Force to Explain the Path of Hurricanes and Snowballs on Carousels

~ Josh Fialkow ~ 1 Comment

For a long time I wondered how we can mathematically intuit the fictitious Coriolis Force (title image taken from en.wikipedia.org/wiki/Coriolis_force) . For those unacquainted with this force, the best place to understand it is on a carousel. While riding this carousel, imagine you are on the southern most spot and your friend is on the … Continue reading Stranger than Fiction: A Derivation of the Coriolis Force to Explain the Path of Hurricanes and Snowballs on Carousels

The Tiniest Changes Add Up: How Infinitesimals Preserve Constant Area in Geometry and Unit Elasticity in Economics

~ Josh Fialkow ~ Leave a comment

In our previous post, we found that area cannot be maintained through equal but opposite percent changes in length and width using algebraic techniques. We determined that when we increased length by a certain percentage and decreased width by the same percentage, our new area calculation was derived through a downward sloping parabola of the … Continue reading The Tiniest Changes Add Up: How Infinitesimals Preserve Constant Area in Geometry and Unit Elasticity in Economics