The Navier-Stokes equations are key in science and engineering, modeling phenomena such as the weather, ocean currents, and airflow around a wing. The existence of smooth solutions to Navier-Stokes remains unsolved, and it's one of the millennium problems.
#MathType
L'Hôpital's Rule was actually devised by Johann Bernoulli. It appeared in the first-ever
#Calculus
textbook by Guillaume de L'Hôpital, thanks to an agreement: Bernoulli would inform L'Hôpital only of his latest findings in exchange for an annual payment of 300 Francs.
#MathType
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 Analysis
#MathType
Today is the 202nd birthday anniversary of Sir George Gabriel Stokes, a big name in both
#Physics
and
#Math
. One of the theorems that bears his name in
#VectorCalculus
is a theorem so powerful that it contains many special cases that are themselves powerful theorems.
#MathType
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.
#MathType
#mathfacts
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.
#MathType
#mathfacts
A staple in Probability and Statistics but also in Machine Learning and Data Science: Bayes' Theorem gives the probability of an event given previous information.
#MathType
Laplace transform converts a function that depends on a real variable t into a function that depends on a complex variable s. This mathematical tool finds widespread application in engineering, particularly in circuit analysis.
#MathType
#math
#mathematics
#mathfacts
A way of computing the n-th Fibonacci number is by brute force (keep adding the last 2 members of the sequence to find a new one).
Another easier and faster way is to use Binet's Formula, which calculates any Fibonacci number exactly using the golden ratio
#NumberTheory
#MathType
Entropy is often said to be the disorder of a system, as it is determined by the number of random microstates. A microstate is a microscopic configuration consistent with the macroscopic quantities that characterize a thermodynamic system.
#MathType
#math
#mathfacts
The Laplace transform, named after Pierre-Simon Laplace, transforms a function of a real variable t (normally time) to a function of a complex variable s (complex frequency). It is often used in engineering, like in circuit analysis.
#MathType
The Laplace transform, named after Pierre-Simon Laplace, transforms a function of a real variable t (normally time) to a function of a complex variable s (complex frequency). It is often used in engineering, like in circuit analysis.
#MathType
The
#binomial
distribution was first studied in the context of games of pure chance, and it's now widely used to analyze data in many fields. It models the number of successes when performing n trials of an experiment with a probability of success p.
#MathType
#math
#mathfacts
The Navier-Stokes equations are key in science and engineering, modeling phenomena such as the weather, ocean currents, and airflow around a wing. The existence of smooth solutions to Navier-Stokes remains unsolved, and it's one of the millennium problems.
#MathType
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.
#MathType
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.
#MathType
#mathfacts
Laplace transform converts a function that depends on a real variable t into a function that depends on a complex variable s. This mathematical tool finds widespread application in engineering, particularly in circuit analysis.
#MathType
#math
#mathematics
#mathfacts
Green's Theorem is one of the big results in Differential Calculus. It relates the integral over a simple closed curve to another over its enclosed area, allowing simple calculation of some otherwise hard integrals. E.g. when dealing with electromagnetism in Physics
#MathType
Laplace transform converts a function that depends on a real variable t into a function that depends on a complex variable s. This mathematical tool finds widespread application in engineering, particularly in circuit analysis.
#MathType
#math
#mathematics
#mathfacts
Siméon-Denis Poisson derived the famous Poisson distribution in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. Today, this probability function models a wide range of events in all kinds of disciplines.
#MathType
Entropy is often said to be the disorder of a system, as it is determined by the number of random microstates. A microstate is a microscopic configuration consistent with the macroscopic quantities that characterize a thermodynamic system.
#MathType
#math
#mathfacts
Suppose that you want to find the maximum or minimum values of a multivariate function, but you also want your solution to satisfy some restrictions... This problem is called "conditional extrema", and some cases can be solved using the method of Lagrange Multipliers.
#MathType
Bernoulli's principle in
#fluid
dynamics relates the height, the pressure, and the speed of the fluid at each point of a flowing stream. It can be used to calculate the lift force on an airfoil, among a wide range of
#engineering
applications.
#MathType
#math
#mathfacts
Suppose that you want to find the maximum or minimum values of a multivariate function, but you also want your solution to satisfy some restrictions... This problem is called "conditional extrema", and some cases can be solved using the method of Lagrange Multipliers.
#MathType
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.
#MathType
#mathfacts
Math fact: both the Jacobian matrix and its determinant are called “The Jacobian”. Confusing right?
The Jacobian matrix of a vector-valued function, named after Carl Gustav Jacobi, is the matrix of all its first-order partial derivatives.
#math
#mathematics
#mathfacts
L'Hôpital's Rule was actually devised by Johann Bernoulli. It appeared in the first ever Calculus textbook, by Guillaume de L'Hôpital, thanks to an agreement: Bernoulli would inform L'Hôpital only of his latest findings in exchange for an annual payment of 300 Francs.
#MathType
The Pearson correlation coefficient reigns as the go-to metric for assessing linear correlations. Raging from –1 to 1, it measures the strength and direction of the relationship between two variables.
#MathType
#math
#mathematics
#mathfacts
Patterns that eventually fail... but take a long time to, like some of the Borwein integrals. This, among others, is why a formal proof is required to label a mathematical idea true or false
#MathType
The Cauchy–Bunyakovsky–Schwarz inequality is one of the most famous and used inequalities in mathematics. Particular cases of it include Cauchy's inequality for the dot product and Holder's inequality for square-integrable functions.
#MathType
Cauchy’s Integral Formula is a central statement in complex analysis. It says that given a holomorphic function f(z) defined on a closed curve C, if we know the values of f(z) on the boundary curve, we know everything about the function inside C.
#MathType
The most probable distance between the nucleus and the electron in a hydrogen atom in its ground state is given by the Bohr Radius. This physical constant is named after Niels Bohr, due to its role in the Bohr model of the atom.
#MathType
The hyperbolic functions are expressed like the ordinary trigonometric functions but without the “i” appearing in the complex exponential and are defined for the hyperbola rather than on the circle. These functions often arise in many problems of mathematics.
#MathType
#math
The central limit theorem states that as the sample size of a random variable increases, the distribution of sample means becomes more and more normal, regardless of the underlying distribution of the population. It is fundamental in probability theory and
#statistics
.
#MathType
L'Hôpital's Rule was actually devised by Johann Bernoulli. It appeared in the first-ever
#Calculus
textbook by Guillaume de L'Hôpital, thanks to an agreement: Bernoulli would inform L'Hôpital only of his latest findings in exchange for an annual payment of 300 Francs.
#MathType
Cauchy’s Integral Formula is a central statement in complex analysis. It says that given a holomorphic function f(z) defined on a closed curve C, if we know the values of f(z) on the boundary curve, we know everything about the function inside C.
#MathType
Math fact: both the Jacobian matrix and its determinant are called “The Jacobian”. Confusing right?
The Jacobian matrix of a vector-valued function, named after Carl Gustav Jacobi, is the matrix of all its first-order partial derivatives.
#math
#mathematics
#mathfacts
Siméon-Denis Poisson derived the famous Poisson distribution in 1830 to describe the number of times a gambler would win a rarely won game of chance in a large number of tries. Today, this probability function models a wide range of events in all kinds of disciplines.
#MathType
What's the relation between the concept of differentiability and the existence of derivatives? Although the concepts are equivalent for functions of one variable, in higher dimensions differentiability is a stronger property!
#MathType
#math
#mathematics
#mathematical
#mathfacts
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.
#math
#mathfacts
Pascal's Combinatorial Rule can be proved in a sentence:
"If from a set of n+1 we choose k objects, we either exclude the last one, in which case we choose k from the first n, or include it, in which case we choose k-1 from the remaining n".
#MathType
#math
#mathfacts
The Taylor series expansion of a function represents it as an infinite sum of terms that depend on the function's derivatives evaluated at a particular point. This allows for a very precise approximation of the function around that point.
#MathType
#math
#mathfacts
The Coriolis effect is due to a fictitious force that acts on objects in motion within a rotating frame of reference. On the Earth, this effect shows as a deflection of the trajectories of objects not connected to the ground.
#MathType
#math
#mathematics
#mathfacts
Laplace transform converts a function that depends on a real variable t into a function that depends on a complex variable s. This mathematical tool finds widespread application in engineering, particularly in circuit analysis.
#MathType
#math
#mathematics
#mathfacts
Today in 1977, Maryam Mirzakhani was born. She was the first woman to win a Fields Medal, and made key contributions to the fields of hyperbolic geometry and Riemann surfaces. In her honour the 12th of May is celebrated as
#WomenInMathDay
.
#MathType
An elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. Elliptic curves can be endowed with a group law on their points, which makes them a very interesting object with special properties.
#MathType
The Taylor series expansion of a function represents it as an infinite sum of terms that depend on the function's derivatives evaluated at a particular point. This allows for a very precise approximation of the function around that point.
#MathType
#math
#mathfacts
The Gamma function was created as a solution to the problem of extending factorials beyond the positive integers. Remarkably, every generation of mathematicians since the 18th century has discovered something about it.
#MathType
#math
#mathematics
#mathfacts
The Mean Value Theorem is a classical elementary result proven by
#Cauchy
in 1823 using the ideas of modern calculus. The core of the theorem was partially known way before though, going back to Parameshvara, an Indian astronomer around the early 15th century.
#Calculus
#MathType
The continuity equation, also called the transport equation, describes the transport of some quantity, and represents stronger, local forms of conservation laws.
#MathType
The central limit theorem states that as the sample size of a random variable increases, the distribution of sample means becomes more and more normal, regardless of the underlying distribution of the population. It is fundamental in probability theory and
#statistics
.
#MathType
Bernoulli's principle in
#fluid
dynamics relates the height, the pressure, and the speed of the fluid at each point of a flowing stream. It can be used to calculate the lift force on an airfoil, among a wide range of
#engineering
applications.
#MathType
#math
#mathfacts
Have you ever heard that the sum of all natural numbers is -1/12? Of course this doesn't make sense in the usual sense of sum, but using a
#summation
method based on analytic continuation of the
#Riemann
Zeta function leads to this result.
#MathType
#math
#mathfacts
For today something very basic: the complex conjugate of a complex number has an equal real part and imaginary part equal in magnitude but opposite in sign. They often appear when finding roots of polynomials.
#MathType
Euler’s identity, beauty in a formula.
Sometimes called "the most beautiful equation in mathematics", Euler's Identity connects together 5 seemingly unrelated mathematical constants: e, π, i, 0, and 1. Do you know the proof of this identity?
#math
#mathematics
#mathfacts
Last but not least, the Third Law of Kepler after having published the others. This one states that: “The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit”.
#MathType
Gaussian elimination is a powerful tool used in various fields of mathematics and beyond. By systematically reducing a matrix to its row echelon form, this algorithm simplifies complex problems and enables us to find solutions.
#math
#mathematics
#mathfacts
Bayes' Theorem is a mathematical formula used in probability theory and
#statistics
to calculate conditional probability. This theorem helps to determine the probability of an event, considering prior knowledge of conditions that may be linked to the event.
#MathType
#math
The mathematician Emmy Noether announced an important theorem in physics that explains the connection between symmetry and conservation laws. It is a profound reinterpretation of the Euler-Lagrange equation.
#MathType
#Stochastic
convergence formalizes the idea that a sequence of unpredictable events can be expected to settle into a pattern. There are several ways in which this can happen, giving rise to the different notions of convergence of
#random
variables.
#MathType
Ramanujan's master theorem is an important tool to calculate definite integrals and infinite series. It provides an analytic expression for the Mellin transform of an analytic function.
#MathType
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.
#MathType
The Coriolis effect is due to a fictitious force that acts on objects in motion within a rotating frame of reference. On the Earth, this effect shows as a deflection of the trajectories of objects not connected to the ground.
#MathType
#math
#mathematics
#mathfacts
Brownian motion is a physical phenomenon in which some quantity is constantly undergoing small and random fluctuations. It is habitually used to describe the random motion of particles suspended in a liquid or gas.
#MathType
The
#binomial
distribution was first studied in the context of games of pure chance, and it's now widely used to analyze data in many fields. It models the number of successes when performing n trials of an experiment with a probability of success p.
#MathType
#math
#mathfacts
An elegant example of the conservation of angular momentum, the rotational equivalent of linear momentum, is when a figure skater does a pirouette. Their rotational speed increases as their moment of inertia decreases by drawing in their arms and legs.
#MathType
#math
#mathfacts
The Pearson correlation coefficient reigns as the go-to metric for assessing linear correlations. Raging from –1 to 1, it measures the strength and direction of the relationship between two variables.
#MathType
#math
#mathematics
#mathfacts
The central limit theorem states that as the sample size of a random variable increases, the distribution of sample means becomes more and more normal, regardless of the underlying distribution of the population. It is fundamental in probability theory and
#statistics
.
#MathType
Have you ever heard that the sum of all natural numbers is -1/12? Of course this doesn't make sense in the usual sense of sum, but using a
#summation
method based on analytic continuation of the
#Riemann
Zeta function leads to this result.
#MathType
#math
#mathfacts
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.
#MathType
#math
#mathematics
#mathfacts
Bayes' Theorem is a mathematical formula used in probability theory and
#statistics
to calculate conditional probability. This theorem helps to determine the probability of an event, considering prior knowledge of conditions that may be linked to the event.
#MathType
#math
Sir George Gabriel Stokes is a big name in both
#Physics
and
#Math
. One of the theorems that bears his name in
#VectorCalculus
is a theorem so powerful that it contains many special cases that are themselves powerful theorems.
#MathType
Here's a memory of high school calculus: Leibniz product rule, which allows to calculate the derivatives of products of functions in terms of the derivatives of each function.
#MathType
Entropy is often said to be the disorder of a system, as it is determined by the number of random microstates. A microstate is a microscopic configuration consistent with the macroscopic quantities that characterize a thermodynamic system.
#MathType
#math
#mathfacts
Kepler’s laws, published by Johannes Kepler, are three scientific laws that describe the motion of planets around the Sun. The first Kepler law states, "The orbit of every planet is an ellipse with the Sun at one of the two foci”.
#MathType
#math
#mathematics
#mathfacts
Lebesgue's dominated convergence theorem provides sufficient conditions under which pointwise convergence of a sequence of functions implies convergence of the integrals. It's one of the reasons that makes
#Lebesgue
integration more powerful than
#Riemann
integration.
#MathType
Kepler's second law states: “A line joining a planet and the Sun sweeps out equal areas during equal intervals of time”. An expression for the area swept out by the radius vector between two-time instants can be derived.
#MathType
#math
#mathematics
#mathfacts
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.
#MathType
An adjacency matrix is a way of representing a finite
#graph
. The matrix elements indicate whether pairs of vertices are adjacent or not in the graph. The eigenvalues and eigenvectors of an adjacency matrix are studied in
#spectral
#GraphTheory
.
#MathType
#math
#mathfacts
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.
#MathType
#math
#mathematics
#mathfacts
Kepler’s laws, published by Johannes Kepler, are three scientific laws that describe the motion of planets around the Sun. The first Kepler law states, "The orbit of every planet is an ellipse with the Sun at one of the two foci”.
#MathType
#math
#mathematics
#mathfacts
A proof that there are infinitely many prime numbers using Riemann's Zeta function and Euler's product formula for it. Do you know any shorter proof?
#MathType
L'Hôpital's Rule was actually devised by Johann Bernoulli. It appeared in the first-ever
#Calculus
textbook by Guillaume de L'Hôpital, thanks to an agreement: Bernoulli would inform L'Hôpital only of his latest findings in exchange for an annual payment of 300 Francs.
#MathType