MathType

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

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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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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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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 Iverson Bracket is a powerful generalization of the well-known Kronecker Delta, very useful for embedding logical conditions and truth values into your formulas

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When paired with the linearity of the integral, Cavalieri's quadrature formula allows computing the integrals of all polynomials.

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The Extreme Value Theorem is a fundamental result in calculus. It guarantees that a continuous function on a closed interval has both a maximum and a minimum value.

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The folium of Descartes is a mathematical curve named after René Descartes. It is a single loop with one node and two asymptotes at the end and symmetrical about the line y=x.

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Convolution in math is an operator applied on two functions which results in another function that reflects how the shape of one is modified by the other. It can also act on periodic functions and be defined for the discrete domain.

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The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be represented uniquely as a product of prime numbers. Without this property, an irreducible number (only divisible by itself or the unit) would not need to be prime.

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Behind every geometric figure lies a fascinating range of formulas.

Can you demonstrate the formula in the picture?
Hint! The dodecahedron comprises 12 pyramids with a pentagon as the base.

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The Binomial Theorem allows to quickly expand expressions in the form (x+y)^n as it describes the algebraic expansion of powers of a binomial.

The expansion will have n+1 terms and there is always a symmetry in the coefficients in front of the terms.

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The AM-GM inequality relates the arithmetic mean (AM) to the geometric mean (GM).

For non-negative real numbers, their geometric mean will never exceed their arithmetic mean, with equality if and only if all chosen numbers are equal.

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In every finite undirected graph, the number of vertices that touch an odd number of edges is even. In a graph G, the sum of the degrees of the vertices is equal to twice the number of edges.

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The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be represented uniquely as a product of prime numbers. Without this property, an irreducible number (only divisible by itself or the unit) would not need to be prime.

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The Extreme Value Theorem is a fundamental result in calculus. It guarantees that a continuous function on a closed interval has both a maximum and a minimum value.

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A Rose or Rhodonea Curve is a curve shaped like a petalled flower. It is a sinusoid specified by the cosine or sine functions with no phase angle that is plotted in polar coordinates.

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Jensen's #inequality generalizes the fact that the secant line of a convex function lies above the graph of the function. Here, we present the finite statement of this inequality.

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It was 1962 when Kenneth E. Iverson first introduced the current notations and names of the floor and ceiling functions. He relied on the previous bracket notation of Gauss and the notion of the integer part of a real number x of Legendre.

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Lami's Theorem is a straightforward application of the sine rule to #Mechanics dating back to the 17th century. This rule is still used in structural analysis in #CivilEngineering to keep those buildings standing!

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The Gelfond-Schneider theorem is a partial positive answer to Hilbert's 7th Problem, giving a check for transcendability (not being the solution to ANY polynomial equation in rational coefficients) for a big class of numbers.

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The equipartition theorem is widely used in classical statistic mechanics, it relates the temperature of a system to the average energy of its particles. It can be used to derive the ideal gas law and to predict the properties of stars.

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Liouville's theorem states that every bounded function that is holomorphic in the whole complex plane must be constant. It is a classical result in #ComplexAnalysis.

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The linear transport equation models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the easiest partial differential equations and one of the few that admits an analytic solution.

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In astrophysics, the free-fall time serves to study the time scales governing star collapse under their own gravitational attraction. For the Sun, for instance, with a mean density of 1.41 g/cm^3, the free-fall time is about 30 min!

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Faraday became one of the most influential scientists in history by establishing the concept of electromagnetic field and discovering many principles related to #Electromagnetism, like his law of electromagnetic induction.

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The Fermi energy, named after the physician Enrico Fermi, is the difference of energy between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions, when the temperature is absolute zero.

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The “sine cardinal” function, commonly known as “sinc” function, often appears in digital signal processing and the theory of Fourier transforms. There are two slightly different widely used definitions for this function.

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