MathType
@MathType
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Following
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Statuses
4,814
Type and handwrite mathematical notation. Easily include quality math equations in your documents and digital content. Part of @wiris_science .
Barcelona, Spain
Joined December 2008
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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
15
246
1K
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
14
201
967
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
9
184
918
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
2
218
905
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
4
195
902
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
8
181
855
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
12
272
793
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
8
214
811
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
0
147
763
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
10
156
756
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
8
150
744
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
11
153
734
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
8
139
710
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
1
153
712
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
4
132
674
The Mean Value Theorem is a classical elementary result proven by
#Cauchy
in 1823 using the ideas of modern calculus.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
2
119
665
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
2
130
651
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
4
112
629
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
3
140
636
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
3
109
629
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
6
135
611
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
2
129
591
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
5
98
584
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
3
127
586
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
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
5
136
574
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
6
111
563
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
6
117
548
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
2
102
542
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
1
103
544
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
3
129
539
Here's Schwartz's Theorem, also known as the "equality of mixed partials". It's one of the classical
#calculus
results, and a relief for students.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
#Analysis
5
93
541
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
3
107
526
The fourth maxwell equation is the generalization of Ampère’s circuital law and can be written both in differential and integral forms.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
3
99
521
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
1
117
536
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
7
129
533
The fundamental lemma of the calculus of variations is a key result in the field of mathematical analysis. It is a simple but profound statement, and its proof involves a series of steps. How would you prove it?
#MathType
#NumberTheory
#math
#mathematics
#mathematical
#science
19
77
530
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
3
105
538
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
4
149
523
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
4
96
519
The Mean Value Theorem is a classical elementary result proven by
#Cauchy
in 1823 using the ideas of modern calculus.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
4
99
517
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
1
107
512
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
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
5
107
503
Here's the famous formula of
#integration
by parts, one of the best-known techniques to perform antiderivatives.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
6
109
499
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
4
95
497
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
2
105
494
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
7
111
494
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
17
116
490
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
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
3
92
490
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
1
127
479
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
2
107
474
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
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
6
102
487
#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
4
108
474
Here's Schwartz's Theorem, also known as the "equality of mixed partials". It's one of the classical
#calculus
results, and a relief for students.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
#Analysis
4
84
469
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
7
102
469
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
0
93
459
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
2
85
453
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
1
83
465
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
4
107
452
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
9
101
455
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
1
96
451
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
5
86
443
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
9
103
452
The continuity equation, also called the transport equation, describes the transport of some quantity, and represents stronger, local forms of conservation laws.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
1
98
443
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
5
110
438
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
2
93
439
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
3
90
437
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
7
76
434
Here's the famous formula of
#integration
by parts, one of the best-known techniques to perform antiderivatives.
#MathType
#math
#mathematics
#mathematical
#mathematician
#mathproblems
#mathfacts
2
75
434
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
3
66
437
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
3
109
444
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
1
89
435
