Category: math

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

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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

Geometrical Proof of the Cross Product

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From our previous discussion (https://mathintuitions.com/2022/09/22/stranger-than-fiction-a-derivation-of-the-coriolis-force-to-explain-the-path-of-hurricanes-and-snowballs-on-carousels/), the cross product of the angular velocity (with components vi and vj) with the position vector was and is the tangential velocity vector. It can also be written more specifically as xivj - yjvi). The cross product of two vectors A and B with 2 dimensions is calculated as AiBj … Continue reading Geometrical Proof of the Cross Product

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