Laws of Math & Physics
Explore 55+ foundational laws and theorems across mathematics and physics. Click any card for an AI-powered explanation.
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Gives the roots of any quadratic equation ax² + bx + c = 0.
Expands any power of a binomial as a polynomial sum.
Links five fundamental constants — e, i, π, 1, and 0 — in one elegant equation.
Connects complex exponentials to trigonometric functions, fundamental to signal processing.
Connects differentiation and integration, showing they are inverse operations.
Differentiates composite functions by multiplying derivatives of outer and inner functions.
Differentiates the product of two functions.
Transforms the integral of a product into a simpler integral.
Approximates any smooth function as an infinite polynomial around a point a.
Guarantees a point c where the instantaneous rate equals the average rate over [a, b].
Evaluates indeterminate limits (0/0 or ∞/∞) using derivatives.
Relates a surface integral of a curl to a line integral around its boundary.
Generalizes the Pythagorean theorem to any triangle.
Relates sides of a triangle to the sines of the opposite angles.
Connects the geometry (curvature) of a surface to its topology (Euler characteristic).
Updates a prior probability given new evidence.
The sample mean of many independent variables converges to a normal distribution.
The bell curve probability density function, ubiquitous in nature and statistics.
A matrix A scales eigenvector v by eigenvalue λ — fundamental to quantum mechanics, ML, and more.
The inner product of two vectors is bounded by the product of their norms.
Decomposes any function into its constituent frequencies.
If p is prime and a is not divisible by p, then aᵖ ≡ a (mod p). Foundation of RSA encryption.
Force equals mass times acceleration — the cornerstone of classical mechanics.
A spring's restoring force is proportional and opposite to its displacement.
Every mass attracts every other mass with a force proportional to their masses and inversely to the square of the distance.
Force directed toward the center required to keep an object moving in a circular path.
The square of an orbit's period is proportional to the cube of its semi-major axis.
Total linear momentum of an isolated system remains constant.
Total mechanical energy of an isolated system is conserved.
In steady fluid flow, pressure plus kinetic and potential energy density is constant.
The equation of motion derived from the Lagrangian L = T - V, generalizing Newton's laws.
Relates pressure, volume, and temperature of an ideal gas.
The change in internal energy equals heat absorbed minus work done by the system.
Entropy of an isolated system never decreases — the arrow of time.
Total power radiated by a black body is proportional to the fourth power of temperature.
Links microscopic disorder (number of microstates Ω) to macroscopic entropy.
Voltage across a conductor equals current times resistance.
Electrostatic force between two point charges is proportional to their charges and inversely to the square of distance.
A changing magnetic flux through a loop induces an electromotive force (EMF).
Force on a charged particle due to electric and magnetic fields.
The total electric flux through any closed surface equals the enclosed charge divided by ε₀.
Speed of light derived from Maxwell's equations as 1/√(μ₀ε₀).
The wave equation governing quantum state evolution over time.
Position and momentum of a particle cannot both be known precisely at the same time.
Every particle has an associated wavelength inversely proportional to its momentum.
Energy of a photon is proportional to its frequency, marking the birth of quantum theory.
The probability of finding a particle at position x is the square of its wave function.
Mass and energy are interchangeable, connected by the square of the speed of light.
A moving clock runs slower than a stationary one — time passes differently at different speeds.
Objects moving at relativistic speeds appear shorter along the direction of motion.
The full relativistic relationship between energy, momentum, and rest mass.
Describes refraction — how light bends when passing between media of different refractive indices.
Observed frequency changes when source or observer moves relative to a wave medium.
Constructive interference occurs when path difference is an integer multiple of wavelength.