The length of a vector, also known as its magnitude or norm, refers to the measure of its size or extent. It provides information about the vector's distance from the origin to its endpoint in a coordinate system.

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The Fresnel equations describe the reflection and transmission of light when it impinges on an interface between different optical media. The equations depend on the incident, reflected and transmission angles and the refractive index of both media.

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Lagrange’s four-square theorem asserts that any positive whole number can be written as the sum of four squares of integers. Leave an example in the comments!

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Friedrich Wilhelm Bessel is particularly known for the second-order Bessel Differential Equation and the Bessel functions, which are important for many problems of wave propagation and static potentials.

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Wilhelm Wirtinger was an Austrian mathematician born in 1865. One of his contributions is Wirtinger's Inequality, widely used in many areas of #Analysis . Notably, he also had some remarkable students like Gödel and Schrödinger.

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Lambert's W function, also called product logarithm, is defined as the inverse of the function f(x)=x e^x. It's a multivalued complex function that can't be expressed in terms of elementary functions. It's useful in #combinatorics and #biochemistry .

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Symmetry, regularity and elegance.

Platonic solids, also known as regular polyhedra, are a special class of three-dimensional geometric shapes. There are exactly five Platonic solids. Which one do you like the most?

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Bertrand's postulate is a theorem that guarantees the existence of at least one prime number p between any given integer n greater than 3 and 2n. It was completely proved by Chebyshev in 1852.

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Another way of computing the n-th Fibonacci number is to use Binet's Formula, which calculates any Fibonacci number exactly using the golden ratio.

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The Pauli matrices are a set of three 2 × 2 complex matrices used in #QuantumMechanics in the context of the intrinsic angular momentum or spin. They appear in Pauli's equation which is the formulation of the Schrödinger equation for spin ½ particles.

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Matrices are fundamental mathematical objects that have numerous applications in various fields. Tell us, what do you use them for?

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The Doppler Effect states that the observed frequency of a wave depends on the relative speed of the source and the observer and the emitted frequency f0. Doppler later attempted to use this concept to explain the color of binary stars.

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Trigonometry is fun, but non-euclidean fun is better! Just remember: the angles of a triangle on a sphere add up to more than 180 degrees and its sides divided by the radius are just angles.

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There is a simple and straightforward formula for calculating the components of the orthogonal projection of a vector. As seen in the picture, it is a simple scalar.

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Absolute continuity has important implications in calculus and analysis, particularly in the exploration of integration and differentiation. Functions that possess absolute continuity exhibit a range of desirable properties. which ones you know?

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The partial differential equation called the heat equation describes how some quantity, such as heat, distributes over time in a homogeneous and isotropic solid medium.

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It is not obvious at all to see that addition works in unexpected ways if we add an infinite number of summands. This fact troubled early 19th-century mathematicians until Riemann came up with an explanation to this bizarre behavior in the 1850s.

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De Gua's Theorem is a three-dimensional analog of the Pythagorean theorem. This theorem offers valuable insights into spatial relationships. Have you heard about it?

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The binomial coefficient is a mathematical expression that quantifies the number of possible combinations when selecting k items from a set of n distinct items without regard to the selection order. It has many applications. Could you name one?

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The Fourier series is a great technique to break any function into simple sine waves, which can then be pieced together for an approximation of arbitrary precision. This is key in any technology involving signal processing of EM waves e.g. radio, wifi.

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