➕Math Lab ➕
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Two pages of Albert Einstein’s 1913 handwritten notes filled with tensor equations, Christoffel symbols, and covariant expressions from his work on gravitation.

These notes come from the period when Einstein collaborated with Marcel Grossmann on the Entwurf theory, an early non-Riemannian metric approach that preceded the final 1915 general relativity field equations.

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We know what it does.
We still don’t know what it is.

Dark energy is one of the biggest mysteries in modern cosmology. The universe is not only expanding, its expansion is accelerating. We call whatever is responsible for this acceleration dark energy, but its true nature remains unknown.

Current observations suggest dark energy makes up roughly 68% of the universe’s total energy content. Possible explanations include a cosmological constant, a changing form of energy, or even a modification of gravity.

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In 1831, Michael Faraday discovered electromagnetic induction by showing that a changing magnetic field could generate an electric current. Joseph Henry independently observed the same phenomenon around the same period.

Faraday’s law captures that result mathematically: changing magnetic flux through a coil induces an electromotive force. More coil turns and faster flux changes produce a larger induced voltage.

Heinrich Lenz later described the direction of the induced current, showing that it acts to oppose the change that produced it. James Clerk Maxwell eventually incorporated Faraday’s discovery into the equations that unified electricity and magnetism.

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A black hole is not defined only by darkness. Its mass sets the scale of its horizon, its quantum temperature, and the amount of entropy it can contain.

For a non-rotating black hole, radius grows with mass while Hawking temperature falls as mass increases. Larger black holes are therefore colder. Entropy grows with the horizon’s area, connecting gravity, quantum theory, and thermodynamics within a single object.

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Kalamsat was the smallest satellite ever launched, which weighed a mere 64 grams and measured 3.8 cm on each side.

It was designed by an 18-year-old Indian student named Rifath Sharook and launched on June 22, 2017, through NASA's sounding rocket from Wallops Island.

It was named after former Indian President Dr. A. P. J. Abdul Kalam. Its main goal was to test the performance of 3D-printed structures in space and measure space radiation. The total suborbital flight lasted 240 minutes.

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FUN FACT

During a lecture Lord Kelvin wrote an integral on the board, turned to the class and asked if they knew what a mathematician is.

"A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician."

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A whip cracks because a rapidly moving loop travels down its tapered length, transferring energy toward a section with progressively less mass.

As the loop approaches the narrow end, its speed rises sharply. Part of the whip can exceed the local speed of sound, creating a shock wave similar to a small sonic boom. The sound comes from compressed air, not from the tip striking anything.

High-speed imaging shows that the fastest point may lie within the moving loop rather than at the absolute tip. Its exact motion depends on the whip’s taper, stiffness, and the way it is swung.

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Jean-Pierre Serre turns 100.

Youngest Fields Medalist ever: 27.
First Abel Prize laureate: 2003.

and still editing Wikipedia.

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An object does not have to move to have energy. Einstein’s equation connects its mass, m, with its rest energy, E₀. The factor c², the speed of light squared, gives the relationship between them. Even when an object is completely at rest, its mass corresponds to an enormous amount of energy.

This connection also changes how we understand the mass of a whole system. Heat a sealed container of gas, and the energy added increases its mass by a tiny amount, even though no particles have been added. The mass of a system reflects its total energy in its rest frame, including the motion and interactions of its constituents.

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The closer you get to the speed of light, the slower time passes for you relative to someone on Earth.

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Quadric Surfaces: Standard Forms and Formulas

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How does a rocket work perfectly well in the vacuum of space?

In a vacuum, there is no air for a rocket to push against in order to move forward.

But a rocket carries its own propellant with it. It accelerates exhaust gases backward at high speed, causing the rocket to accelerate forward. This follows the principle of conservation of momentum: as the exhaust gains momentum backward, the rocket gains momentum forward

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A passing siren can change pitch while emitting a steady tone. As the source approaches, successive wavefronts arrive closer together in time, raising the frequency heard. As it recedes, they arrive farther apart, lowering it.

The Doppler effect turns wave timing into a measure of motion. Radar uses frequency shifts in reflected waves to measure speed toward or away from the detector.

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These wells illustrate how gravity becomes more extreme near compact objects.

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The 2026 Nobel Prize in Physics has been awarded to Francis Halzen for his decisive contributions to the IceCube Neutrino Observatory and the discovery of high-energy neutrinos of astrophysical origin.

So what exactly did Halzen’s Nobel-winning work reveal?

Not neutrinos. They were discovered decades ago.

The breakthrough was finding high-energy neutrinos coming from beyond our Solar System and using them as a new way to study the most violent places in the universe.

Neutrinos are almost impossible to detect. Trillions pass through your body every second, barely interacting with matter. But because they are electrically neutral, they can travel across the universe without being deflected by magnetic fields.

That makes them extraordinary cosmic messengers.

In 1988, Halzen proposed using the Antarctic ice as a gigantic neutrino detector. His idea eventually became IceCube, a cubic kilometre of ice containing thousands of light sensors.

When a neutrino finally collides with matter in the ice, it can produce a tiny flash of Cherenkov light. IceCube detects that flash and reconstructs where the neutrino came from and how much energy it carried.

In 2013, IceCube announced the first compelling evidence for a population of high-energy neutrinos originating far beyond our Solar System.

Then came another remarkable result. In 2017, an IceCube neutrino helped astronomers identify a distant blazar as a likely source of high-energy neutrinos and cosmic rays.

Halzen’s work effectively opened a new window onto the universe. Instead of observing the cosmos only through light, we could now use neutrinos to investigate some of its most extreme environments. A cubic kilometre of Antarctic ice became a telescope for the most elusive particles in nature.

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