➕Math Lab ➕

These wells illustrate how gravity becomes more extreme near compact objects. ✅ Follow @math

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. ✅ Follow @math

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 ✅ Follow @math

Quadric Surfaces: Standard Forms and Formulas ✅ Follow @math

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

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. ✅ Follow @math

Jean-Pierre Serre turns 100. Youngest Fields Medalist ever: 27. First Abel Prize laureate: 2003. and still editing Wikipedia. ✅ Follow @math

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. ✅ Follow @math

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." ✅ Follow @math

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. ✅ Follow @math

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. ✅ Follow @math

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. ✅ Follow @math

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. ✅ Follow @math

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. ✅ Follow @math

Quantum computing replaces the binary logic of classical bits with quantum states. Qubits can exist in superpositions, gates manipulate their amplitudes, and entanglement links their states. Carefully designed interference changes the probability of different outcomes. Measurement then turns the quantum state into classical information. ✅ Follow @math

Transformers come in many forms, each built for a different job. Step-up and step-down transformers change voltage, isolation transformers electrically separate circuits, instrument transformers enable safe measurement, while single-phase and three-phase designs serve different power systems. Ferrite and toroidal cores are optimized for different operating conditions, from high-frequency electronics to efficient power conversion. ✅ Follow @math

The Bohr model gave us a useful picture of the atom: electrons occupying specific quantized energy levels around the nucleus. But the modern quantum picture is very different. An electron is not a tiny particle following a definite orbit like a planet around the Sun. Its state is described by a wavefunction, and the electron cloud represents the probability distribution for where a measurement may find it. That shift was one of the biggest conceptual changes in physics. Instead of asking “What path does the electron take?”, quantum mechanics asks “What state is the electron in, and what are the probabilities of different measurement outcomes?” The Bohr model still matters because it gets an important piece right: atomic energies are quantized. For hydrogen, its predictions for the energy levels are remarkably accurate. But the full quantum-mechanical description replaces fixed orbits with orbitals and wavefunctions. ✅ Follow @math

Srinivasa Ramanujan failed to make it through college, but mathematics was not the problem. In 1904, he entered Government College in Kumbakonam on a scholarship. He became so absorbed in mathematics that he neglected his other subjects. He failed his examinations except mathematics and lost the scholarship. He later tried again at Pachaiyappa’s College in Madras. In the First Arts examination, he passed mathematics but failed all his other subjects. Ramanujan left college without a degree and continued studying mathematics independently. Years later, the same mathematical work brought him to Cambridge and eventually made him a Fellow of the Royal Society. ✅ Follow @math

The universe has been changing for 13.8 billion years, from an unimaginably hot early state to the galaxies, stars, planets, and life we see today. This timeline follows some of the biggest transitions along the way: ✅ Follow @math

Black surfaces absorb more sunlight because they give fewer wavelengths a way back out. White surfaces do the opposite. The colors we see are simply the wavelengths that escaped. ✅ Follow @math