Restored WWII Combat: The Pacific War. Fighting The Kamikaze.
The brutal clash between US Navy fleets and the Japanese Kamikaze. This restored 3-minute trailer captures the terrifying reality of aerial combat in 1944 and 1945.
The Biot-Savart Law is the mathematical core building block of magnetostatics. It calculates the magnetic field (dB) generated by a steady electric current (I) flowing through an infinitesimal conductor length (dL).
Key components:
> Directly proportional to current and length.
> Inversely proportional to the square of the distance (r²).
> Vector Cross Product: Field direction is perpendicular to both the current and the distance vector.
Essential for engineering solenoids, motors, and MRI machines.
The Helmholtz decomposition is one of the fundamental results of vector calculus.
It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.
Bose-Einstein Statistics:
nᵢ = gᵢ / (e^( (εᵢ − μ)/kT ) − 1)
Bose & Einstein’s formula for bosons (integer spin particles that can share the same quantum state). Gives average number of particles nᵢ in energy level εᵢ. Essential for Bose-Einstein condensates, superfluidity & lasers!
The most beautiful equations in physics, all in one stunning grid.
From Newton’s laws that first explained how the world really moves, to Einstein bending spacetime itself, Schrödinger’s quantum waves, the Schwarzschild radius defining the inescapable event horizons of black holes, and the Friedmann equation that charts the birth and expansion of the entire cosmos — these nine formulas are some of humanity’s greatest intellectual triumphs.
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Wtf is going on hahahaha.
Vector fields are actually awesome!!
I feel like my brain is expanding every day by learning Multivariable Calculus. It’s literally anti brain-rot.
This is crazy. And now I can understand and explain these things.
Awesome 🔥🔥🔥
The Fundamental Theorem of Galois Theory ✍️
It connects fields and groups in a remarkable way. Imagine a large Galois extension L over a base field K, along with its Galois group G. The theorem states that there is a clear one-to-one relationship between all the fields that lie between K and L and all the subgroups of G. Each intermediate field M corresponds to the subgroup of automorphisms that keep everything in M unchanged. Conversely, each subgroup H connects to the subfield made up of everything in L that remains fixed by all maps in H. This matching reverses the order: larger fields correspond to smaller subgroups, while larger subgroups correspond to smaller fields. The degree of the extension from M to L equals the size of its subgroup, while the degree from K to M represents how many cosets that subgroup has within G. If an intermediate field is normal over K, its subgroup is normal in G, and the Galois group of that smaller extension is the quotient group. In simple terms, the theorem simplifies complex issues about fields into more manageable questions about the subgroups of a finite group, making it one of the most effective tools in algebra.