Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at about  \pi or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of \pi (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :


The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :


Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:


The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :


The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of \pi^{\pi} to 1000 decimal digits is equal to:


Two expressions involving π and infinite sums:

pi infinite cums

Representation of π in continued fraction form:

pi continued frac form

The sum of π and e, the base of natural logarithms, is equal to:

\pi + e=\displaystyle  \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}

The value of π can be deduced from the identity above:

\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)


Concerning the relationship between science and philosophy

This post consists of the  elements of an answer I wrote at ; the question there was: “Is philosophy the top of all kinds of sciences?”

I think it would be convenient to distinguish between the general term “science”, referring to the state or fact of knowing, or to knowledge acquired by study and learning, and the modern meaning of “science”, mostly referring to mathematics and to the exact sciences using the rules of the scientific method (astronomy, physics…).

Philosophy and science were not separate in Antiquity.

In the original sense, philosophy meant the love, study, or pursuit of wisdom, or the knowledge of things and their causes, theoretical as well as practical.

Pythagoras was a mathematician, and at the same time it is said that he was the first one to call himself a philosopher, or “lover of wisdom”.

Plato was a philosopher who recommended the knowledge and the study of geometry. In The Republic, Plato thought that the best ruler was the king-philosopher.

Aristotle studied nature and wrote works about physics, biology, logic, etc, from a philosophical point of view.

According to the OED:

   “In the Middle Ages, ‘the seven (liberal) sciences’ was often used synonymously with ‘the seven liberal arts’, for the   group of studies comprised by the Trivium (Grammar, Logic, Rhetoric) and the Quadrivium (Arithmetic, Music, Geometry, Astronomy).”

The expression Natural philosophy was frequently used for centuries :

   “Natural philosophy or philosophy of nature (from Latin philosophia naturalis) was the philosophical study of nature and the physical universe that was dominant before the development of modern science. It is considered to be the precursor of natural science.

    From the ancient world, starting with Aristotle, to the 19th century, the term “natural philosophy” was the common term used to describe the practice of studying nature. It was in the 19th century that the concept of “science” received its modern shape with new titles emerging such as “biology” and “biologist”, “physics” and “physicist” among other technical fields and titles; institutions and communities were founded, and unprecedented applications to and interactions with other aspects of society and culture occurred. Isaac Newton‘s book Philosophiae Naturalis Principia Mathematica (1687), whose title translates to “Mathematical Principles of Natural Philosophy”, reflects the then-current use of the words “natural philosophy”, akin to “systematic study of nature”. Even in the 19th century, a treatise by Lord Kelvin and Peter Guthrie Tait, which helped define much of modern physics, was titled Treatise on Natural Philosophy (1867).

In the last few centuries, alchemy separated from chemistry, astrology separated from astronomy, and there was also a certain separation between philosophy on one side, and mathematics and the exact sciences on the other side.

Mathematics became progressively the most prominent and the essential scientific discipline, it is acknowledged as the language of science and of the physical world.

Philosophy is nowadays often regarded as a reflection, view or study of the general principles of a particular branch of knowledge, or activity. There is a philosophy of science, philosophy of mathematics, philosophy of education ,etc.

Some theories or views related to epistemology (which is concerned with the general theory and the study of knowledge) and philosophy, such as rationalism, empiricism, and positivism, share a number of principles with the scientific approach to events and phenomena.

photo of Kant

(Source of the image above: Wikimedia Commons)

A scientist or a physicist can also be a philosopher. Important thinkers can be philosophers and create philosophical systems, but modern philosophers must take into account the advances, discoveries and theories in modern science. A historical example would be Immanuel Kant elaborating his philosophical system and philosophical ideas at the end of the eighteenth century in light of and in relation to the exact sciences known at that time, especially Euclidean geometry and Newtonian classical physics and mechanics.

Some notes about the possibility of a mathematical theory of History

I will present some general remarks and some personal opinions and findings (with a constant concern for accuracy and objectivity) about the attempts at mathematizing History , historical events , processes and phenomena.

My many readings (see my various book pages and the books I have read) and my analysis of History made me realize that important historical events and phenomena are (highly) periodic , and that exact correspondences (or similarities or “homologies” , one of these terms could be chosen, used  and defined) can be found between historical events separated by definite periods of time .

The history of humanity can be considered as the result of the interactions between the lives and actions of human beings  moving and acting in time. All humans have a role to play in the unfolding of historical events , but great men (and women) and great thinkers/scientists/reformers constitute the main group of humans who change and drive historical phenomena and happenings across cultures , nations and empires.

Evolution and progress take place in human history , such as technological/scientific progress, the increase in the global human population over millennia and the increase in the surface of political entities, from city-states to nation-states and to bigger entities , etc, but there are also general principles and definite periodicities and regularities in world history. Among these regularities are the stages or phases of gradual growth and decline through which most great powers and empires pass as they rise/fall and go up and down in time. Certain essential periodicities or cycles in history are accompanied by a change or transmutation in values (“moral” , behavioral , sexual, etc).
Relevant concepts can be defined such as the notion of bi-millenary (exact) correspondence and of bi-millenary periodical return of historical events. I will explain and clarify these concepts more when I have time in future writings.

Some philosophical and religious theories talk about eternal return/recurrence and cyclical time (the notion of eternal return constantly permeates the philosophy of Nietzsche), but these notions are not defined in a precise or scientific way.

I think that Mathematics and the scientific method , i.e observing phenomena and collecting data, creating hypotheses with the adequate mathematical model, experimental/empirical verification of these hypotheses and building  coherent theories, can be used in and applied to human history, provided that this is done in the proper and correct way. Historical chronology plays an important role, and the chronology of events before the Common or Christian era ought to be revised and corrected.

Another prerequisite for the impartial and objective study of history is to abandon preconceived ideas and to have a global perspective of human history , avoiding euro-centrism , afro-centrism and all kinds of ethnocentrisms , and avoiding to get stuck in certain habits such as classifying people and cultures into Western and non-Western. One should take into account the fact that geopolitical groupings and alliances change with the passing of time , centuries and decades.

A new discipline called Cliodynamics was created in the last decade . It is an area of research using mathematical , quantitative approaches and modelings to explain historical processes and societies. The practitioners of this discipline have made some interesting observations about historical events and have tried to formulate mathematically backed theories to interpret historical facts , however I think they have not found or discovered the right , convenient , correct and/or precise way to mathematize History and historical phenomena .

If the right mathematical theory of human history can be elaborated, using the scientific method and testing hypotheses in history could be equivalent to and lead to the (precise) prediction of important events taking place in the future of humankind.

Books about mathematics regarded as noteworthy classics

This post is taken (with some modifications from an answer I gave at  .

Classic books or classics may refer to great and historically important books , or to books widely popular , read and used ,or both . I will try to mention both types of books.
I will cite first a number of books which I think are of primary importance in the history of mathematics , and therefore are generally regarded as classics . This is not a exhaustive list .
Let’s start with some books from Antiquity :
Moving a few centuries later to modern times :
Some recent modern and well known math books that may be regarded as classics :
See also the following links :

The Lagrangian of the Standard Model of particle physics

I will present some notes and explanations related to the Standard Model of particle physics and its Lagrangian . The text in this post is inspired from two answers I gave at .

The Standard model and its Lagrangian form a vast topic . I will attempt to give relevant and accurate information about it.

The story of the Standard Model started in the 1960s with the elaboration of the theory of quarks and leptons  , and continued for about five decades until the discovery of the Higgs boson in 2012.
For a timeline of the history of the Standard Model see the Modern Particle Theory timeline .
The formulation of the Lagrangian of the Standard Model with its different terms and parts mirrored the theoretical and experimental advances associated with particle physics and with the Standard Model.

The Lagrangian function or Lagrangian formalism is an important tool used to depict many physical systems and used in Quantum Field Theory . It has the action principle at its basis .

In simple cases the Lagrangian essentially expresses the difference between the kinetic energy and the potential energy of a system .

The Standard Model of particle physics describes and explains the interactions between the essential components and the fundamental particles of matter , under the effect of the four fundamental forces: the electromagnetic force , the gravitational force , the strong nuclear force , and the weak (nuclear) force.
However , the Standard model is mainly a theory about three fundamental interactions , it does not fully include or explain gravitation .
The Standard model (or SM) is  a gauge theory representing fundamental interactions as changes in a Lagrangian function of quantum fields.  It depicts spinless , spin-(1/2) and spin-1 fields interacting with one another in a way governed by the Lagrangian which is unchanged by Lorentz transformations.

The Lagrangian density or simply Lagrangian of the Standard Model contains kinetic terms , coupling and interaction terms (electroweak and quantum chromodynamics sectors) related to the gauge symmetries of the force carriers (i.e. of the elementary and fundamental particles which carry the four fundamental interactions) , mass terms , and the Higgs mechanism term .

Explicitly , the parts forming the entire Lagrangian generally consist of :
Free fields : massive vector bosons , photons , and leptons.
Fermion fields describing matter.
The Lepton-boson interaction.
Third-order and fourth order interactions of vector bosons.
The Higgs section.

Leptons are the elementary particles not taking part in strong interactions.
All leptons are fermions. They include the electron , muon , and tauon , and the electron neutrino , muon neutrino , and tauon neutrino.
All leptons are color singlets , and all quarks are color triplets.

In the Standard model , the Higgs mechanism provides an explanation for the generation of the masses of the gauge bosons via electroweak symmetry breaking.

Different reference works , books , e-books or textbooks use different or slightly different notations and symbols to describe or designate the entities and terms within the Lagrangian of the SM .

Below is a detailed image of the Lagrangian of the Standard Model  (Source: ).
However I have rearranged it and modified it with the help of Photoshop to make it look more presentable and more readable.

lagrangian of the standard model

The Lagrangian function in the Standard Model , as in other gauge theories , is a function of the field variables and of their derivatives.

G_ {\mu \nu} is the gauge field strength of the strong SU(3) gauge field.
Gluons are the eight spin-one particles associated with SU(3).
A particle which couples to the gluons and transforms under SU(3) is called ‘colored’ or ‘carrying color’.
Gluons and quarks are confined in hadrons.

W_ {\mu \nu} is the gauge field strength of the weak isospin SU(2) gauge field .

The field strength tensor W_ {\mu \nu} is given by :

field strength tensor

where g_2 is the electroweak coupling constant , a dimensionless parameter.

The charged W^+ and W^- bosons and the neutral Z boson represent the quanta of the weak interaction fields between fermions , they were discovered in 1983 .

B_ {\mu \nu} is the gauge field strength of the weak hypercharge U(1) gauge field.
The field strength tensor B_ {\mu \nu}  is given by :

B_ {\mu \nu} = \frac {\partial B_ {\nu}} {\partial\mu} - \frac {\partial B_ {\mu}} {\partial\nu}

In the Standard model , electrons and the other fermions are depicted by spinor fields .
The group U(1) is the set of one-dimensional unitary complex matrices .
U(1) represents the symmetry of a circle unchanged by rotations in a plane.

SU(2) is called ‘the special unitary group of rank two’. It is a non commutative group related to SO(3) , the sphere symmetry in 3 dimensions.
SU(2) is the set of two-dimensional complex unitary matrices with unit determinant.

SU(3) , the special unitary group of rank three , is used in quantum chromodynamics (QCD) .
SU(3) is the set of three-dimensional complex unitary matrices with determinant equal to 1 .
The natural representation of SU(3) is that of 3×3 matrices acting on complex 3D vectors.
The generators of the group SU(3) are eight 3×3 , linearly independent , Hermitian , traceless matrices called the Gell-Mann matrices . These generators can be created from Pauli spin matrices (which are used with the group SU(2) ) .

The SM Lagrangian displays invariance under SU(3) gauge transformations for strong interactions , and under SU(2)xU(1) gauge transformations for electroweak interactions.

The electromagnetic group is not directly the U(1) weak hypercharge group component of the standard model gauge group. The electric charge is not one of the basic charges carried by particles under the unitary product group SU(3)xSU(2)xU(1) , it is a derived quantity.

All the masses vanish in the absence of the Lagrangian term related to the Higgs , due to the invariance of SU(3)xSU(2)xU(1) .

In some texts the gauged symmetry group of the SM is written with subscripts such as:
\text {SU} _c (3)\times\text {SU} _L (2)\times U_Y (1)
In the notation above , the subscript ‘c’ denotes color.
The subscript ‘L’ denotes left-handed fermions.
The subscript ‘Y ‘  distinguishes the group related to the quantum number  of weak hypercharge , expressed by the letter Y , from the group associated with ordinary electric charge, expressed by Q .
U_ {\text {em}} (1) denotes the electromagnetic group.

The Higgs field in the Standard model is a complex scalar doublet. It is generally represented by :

 Higgs field doublet

In the image of the SM Lagrangian above , the Higgs field has the form

Higgs field form

The field h(x) is real .

In the SM Lagrangian image above , \phi _ 0  is equal to v .

As an additional note , the  equation of the Lagrangian  is usually made of a definite number of terms and Lagrangians.
In order to make such an equation look less like a big behemoth and make it more compact ,  it would be simpler to view it or write it first as the sum of Lagrangians :


 or equivalently :

\mathcal {L} = \sum _ {i = 1}^n\mathcal {L} _i

Then each Lagrangian in the equation could be expanded and explained.

Some helpful resources about the Standard Model and its Lagrangian :
Standard Model

The Standard Model of Particle Physics

Standard Model (mathematical formulation)

Gauge Theory of Weak Interactions: Walter Greiner, Berndt Müller: 9783540878421: Books

The Structure and Interpretation of the Standard Model, Volume 2 (Philosophy and Foundations of Physics): Gordon McCabe: 9780444531124: Books

The Standard Model: A Primer: Cliff Burgess, Guy Moore: 9781107404267: Books

An Introduction to the Standard Model of Particle Physics: W. N. Cottingham, D. A. Greenwood: 9780521852494: Books

The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics: Robert Oerter: 9780452287860: Books

Here is also a link to one of the  important papers in the history of the Standard Model written in 1967 by Weinberg and entitled ‘A Model of Leptons’ :

Some algorithmic texture generation with Mathematica

The LineIntegralConvolutionPlot[] function in Mathematica is defined by the Wolfram Mathematica Documentation Center as :

generates a line integral convolution plot of white noise with the vector field {vx,vy}.

LineIntegralConvolutionPlot[] can also generate the plot of an image convolved with a vector field .

I will give some plots of line integral convolutions of a number of vector fields . These plots have often visually appealing forms.

Here is the first plot :

line integral convolution

The Mathematica code for the image above is :

Mathematica code 1

Below are some more line integral convolution plots using various “ColorFunction” options .The frame has been removed from these plots :

This slideshow requires JavaScript.

And below is another line integral convolution plot with the external frame modified with Photoshop . Looks like a nice piece of art … 

plot with frame


The Mathematica code for the image above is :

code for plot


A last example of a line integral convolution plot with a large size :


Mathematica code for the image above ;


For more information about this subject see for example the Wikipedia article about  Line Integral Covolution .

About the definition and the branches of mathematics , and some quotes .

The material for this post is taken or inspired from two answers I wrote at about mathematics.

First I’ll start with a general definition of mathematics taken from the Encyclopedia Britannica (click here to view the online source ) :

“Mathematics, the science of structure, order, and relation that has evolved from elemental practices
of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and
quantitative calculation, and its development has involved an increasing degree of idealization and
abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable
adjunct to the physical sciences and technology, and in more recent times it has assumed a similar
role in the quantitative aspects of the life sciences.”

There are numerous (a word related to number , and hence to mathematics) branches and areas in mathematics , and they are all related and interconnected in one way or another.
Branches can also be divided into sub-branches .

The image below is a combination of (mathematical) images , it was made with Photoshop.

math images

Here are a few examples of mathematical branches or fields of study :
Some would say that the main branches of mathematics are algebra , geometry , and arithmetic (the science of numbers was an important branch of mathematics until the Middle Ages) , others would add analysis , logic , etc.

Algebra generally comprises abstract algebra , the study of rings and fields , linear algebra , and vector spaces. Arithmetic and number theory are viewed as sub-fields of algebra , as well as group theory , representation theory (the study of the elements of algebraic structures as transformations of vector spaces) , homological algebra , commutative algebra …

Analysis involves infinitesimal quantities and the passage to a limit . It comprises (integral and differential ) calculus , vector analysis , complex analysis , real analysis , Fourier analysis , the study of differential equations …

Geometry , the study of space , shapes , and sizes , includes differential geometry , algebraic geometry , analytic geometry , affine geometry , projective geometry , Euclidian and non-Euclidian geometry , noncommutative geometry , topology , etc.

The theory of the foundations of mathematics includes the study of mathematical and symbolic logic , set theory , model theory , proof theory ,…

Pure mathematics is sometimes distinguished from applied mathematics , which comprises disciplines such as probability theory and statistics , numerical analysis , computer algebra , …

Other fields of mathematics include combinatorics , graph theory , ergodic theory , etc.

All the branches of mathematics are used in various ways by the exact sciences  (physics , astronomy , …) and by the social sciences. There is by the way an area of mathematics called ‘mathematical physics’ .

A certain field of mathematics is sometimes interconnected with a number of other fields of mathematics. For example , tensor analysis is closely related to differential geometry , multilinear algebra , mathematical physics , algebraic topology , …

Below are some useful links about the branches of mathematics :

Branches of Mathematics

Areas of mathematics

Glossary of areas of mathematics

Next I give some revealing quotes , opinions and definitions  about mathematics:

  • “Mathematics is the most beautiful and most powerful creation of the human spirit.”
    Stefan Banach.
  • “Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”
    Galileo Galilei .
  • “Neglect of mathematics work injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.”
    Roger Bacon.
  • “Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.”
    David Hilbert .
  • “Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.”
    Deepak Chopra .
  • “Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley.”
    Yōko Ogawa
  • “[When asked why are numbers beautiful?] It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”
    Paul Erdős .
  • “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
    G.H.Hardy .
  • “Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.”
    Felix Klein .
  • “But in my opinion, all things in nature occur mathematically.”
    René Descartes .
  • “The study of mathematics is apt to commence in disappointment… We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the efforts of our mental weapons to grasp it.”
    Alfred North Whitehead .
  • “Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.”
    E.T.Bell .
  • “Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.”
    Bertrand Russell .
  • “The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.”
    James Joseph Sylvester .
  • “Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gaveit sense and you still don’t understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”
    Paul Lockhart .
  • “[Replying to G. H. Hardy’s suggestion that the number of a taxi (1729) was ‘dull’, showing off his spontaneous mathematical genius] No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.”
    Srinivasa Ramanujan .
  • “There cannot be a language more universal and more simple, more free from errors and obscurities…more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes”
    Joseph Fourier .
  • “Humans are like Variables in mathematics, some Dependent, some Independent. Variables are in relationship but remain Variable. Of course, there are some Constants too both in mathematics and humans. Constants help define precisely the relationship between variables. Maybe, that is why humans keep adding (to problems), subtracting (from happiness), multiplying (what else, we are all over earth) and dividing (the earth among themselves).”
    R.N.Prasher .

These are eloquent quotes , not easy  to add to them.
The most important language one could know is not English or French or Spanish (these are of course necessary and useful) , it’s math.

Combined coherently with the rules of the scientific method  , mathematics can be used efficiently to explain and explore  the Natural and Physical World.
Applied adequately to the social sciences , mathematics can inseminate them and bring the human sciences closer and closer to the exact sciences.

One should never underestimate the power of mathematics , reasoning , and exact science.

Related online sources :

Quotes about mathematics from goodreads 

Mathematics Quotes at BrainyQuote



The equations of electrodynamics , tensors , and gravitation-Second Part

I’ll start this continuation of the topic of a previous post by  an exposition and explanation of the concept of tensors in relativity physics  given by Albert Einstein , from his 1916 paper The Foundation of the General Theory of Relativity :

” The laws of physics must be of  such a nature  that  they apply  to systems of reference in any kind of motion.

The general laws of nature are to be expressed by equations which hold good  for all systems of coordinates, that is, are co-variant  with respect to any substitutions whatever (generally covariant).

Let certain things (” tensors “)  be defined with respect to any  system of  coordinates  by  a  number of functions of  the  coordinates , called  the  ” components ”  of the  tensor. There  are  then   certain  rules  by  which   these components can  be calculated   for  a new system of  coordinates , if they are known  for the original  system  of coordinates, and if  the  transformation  connecting  the two systems is known. The things  hereafter  called    tensors are further characterized by the fact  that the equations of transformation for their  components are linear  and  homogeneous.  Accordingly,  all  the components in  the  new system  vanish,  if  they all  vanish in the  original system. If,  therefore, a law  of nature is expressed  by equating all the components of a tensor to zero, it  is  generally covariant. By  examining the  laws of  the formation of  tensors, we acquire  the  means  of  formulating generally covariant laws. “

Source : The Principle of Relativity (Dover Books on Physics)
Einstein’s paper can also be found here:

The Foundation of the Generalised Theory of Relativity

I will deal in this post with the derivation of the Einstein field equations of General Relativity and with the related equations of motion.

As in the previous post , I will not show all the long and detailed calculations but I will give an overview of the equations , calculations and derivations .

We start with the derivation of the field equations free from the sources generating the field.The equations are deduced from the principle of least action.

A small remark : in the equations below , the symbol g written in two ways is the same letter in two different fonts , it represents either the metric tensor (with subscript or superscript) or the determinant of its matrix representation.

The action of the gravitational field is given by:

action gravitation

where χ is the gravitational constant appearing in Einstein’s field equations.
Ω is the product of the differentials of the coordinates.
In four-dimensional space we have :

d \Omega =\text{dx}^0 \text{dx}^1 \text{dx}^2 \text{dx}^3 

In the action integral above , The scalar curvature (or the Ricci scalar) R is related to the Ricci tensor by the equation R = gij Rij  .
And the Ricci tensor is related to the Riemann curvature tensor by the equation:

Ricci tensor

Hence we have:

gravitation action integral

The variation of the action yields:

variation of action

After calculation the first integral in the variation of the action above gives:

first integral in action

The third integral in the variation of the action is equal to zero:

third integral in action

And the variation of the action can be written as:


The variations δgij are arbitrary and the Einstein field equations independent of the sources generating the field
(  Rij – (1/2)gij R = 0 ) can be deduced from the integral above.

The action for matter and the electromagnetic field is given by:

action for matter

This action and its variation involve the Lagrangian density:

action variation

After integrating and calculating we get:

action calculation

Defining the energy momentum tensor (or stress–energy–momentum tensor ) by the relation:

energy momentum tensor

We get for the variation of the action:

action with stress energy tensot

The preceding results lead to the condition δ(Sg+Sme)=0 being written as:

action variation result

The equality above must remain valid for arbitrary variations of δgij  ,and we deduce :

field equations

which are the Einstein field equations of General Relativity.

The contravariant form of the energy-momentum tensor is given by the transformation :

contravariant energy momentum tensor

Below is a schematic description of the contravariant components of the energy-momentum tensor (image from Wikipedia) :

components of contravariant tensor

Note that of we take the more general form of the action :

action with cosmological constant

The field equations for the most general case would become:

field equations with cosmological constant

where λ is the cosmological constant .

The Einstein field equations are in agreement with the conservation of the energy-momentum tensor , which means its divergence is null:

null divergence

The energy-momentum tensor (mixed tensor) can be written as the sum of two terms:

The first one is of electromagnetic origin:

electromagnetic mixed tensor

The second one involves the presence of matter:

mixed tensor matter

The condition of null divergence can be written as:

null divergence two terms

After some calculations we get for the first term :

first term of divergence

And for the second term in the null divergence equation:

second term null divergence

By using the last two equations above , equation (1) can be written as:


which is the equation of motion of a continuous distribution of charge inside a gravitational field and an electromagnetic field.

The references and sources in my Science books page and the problem solvers page can be viewed for books which I have read , worked out or consulted and which are related to the topics in this post.
I found the book by Jean-Claude Boudenot ( Electromagnetisme et gravitation relativistes ) helpful while I was studying these topics in the past.

The equations of electrodynamics , tensors , and gravitation-First Part

The featured image above was made with Mathematica . It represents  the propagation of an electromagnetic wave.

Elements of this post were written for an answer I gave at , the question was about Maxwell’s equations
in tensor form. I have modified the answer and added material and equations related to General Relativity and gravitation. 

Maxwell’s equations are the fundamental equations of classical electromagnetism and electrodynamics. They  can be stated in integral form , in differential form (a set of partial differential equations) , and in tensor form.

The conventional differential  formulation of Maxwell’s equations in the International System of Units is given by:

Maxwell's equations differential form

In the above equations , E is the electric field (vector field) , B is the magnetic field ( pseudovector field) ,  ρ is the charge density , and  J is the current density.
ε0 is the Vacuum permittivity ,
and μ0 is the Vacuum permeability .

The second and third equations above form the first group of Maxwell’s equations (the generalized Faraday’s law of induction and Gauss’s law for magnetism) , and the the first and last equations above form the second group  (Gauss’s law and Ampere’s circuit law extended by Maxwell) .

The equations of electromagnetism (Maxwell’s equations and the Lorentz force) in covariant form (invariant under Lorentz transformations) can be deduced from the Principle of least action .

The electromagnetic wave equation can be derived from Maxwell’s equations , its solutions are electromagnetic (sinusoidal) waves.

All the steps and equations related to the topic of this post will not be shown here because they would  make the post much too long , but an overview of the equations , calculations and derivations  will be presented.

The development of the components of the Lorentz force and (after some calculations) its formulation in tensor form allows the introduction of the electromagnetic field tensor Fij :

Lorentz force

where uj is the four-velocity four-vector .

The variation of the action δS with respect to the coordinates of a particle gives the equations of motion. 

The action for a charged particle in an electromagnetic field can be expressed as:

action variation
q is the electric charge.

The variation of the action gives:

variation of action

After some calculations and an integration by parts the variation of the action becomes:

variation of action and integral

In the equation above , ui is the four-velocity four-vector.

Since the trajectories of the particle are supposed to have the same initial and final coordinates , the first term in the right-hand side of the equality above is equal to zero.
The potentials are a function of the coordinates , and  the following equalities can be used:

potential variation

Which gives for the variation of the action:

variation action & potential

Thus the electromagnetic field tensor emerges  from the variation above ; in fact the electromagnetic tensor is given by the definition:

electromagnetic tensor


Ai is the electromagnetic four-potential comprising the electric scalar potential V and the magnetic vector potential A .

The electromagnetic tensor is antisymmetric:

antisymmetric tensor

And the diagonal components in it are equal to zero:

diagonal components


The components of this tensor can be found using:

potential and field definition

For example:

tensor components

The others components can also be calculated and one gets for the covariant electromagnetic tensor:

electromagnetic tensor components

The components of the contravariant tensor can be found using:

contravariant tensor

where g^{\text{ij}} is the Metric tensor.

The transformation laws for the electromagnetic tensor are given by the general relation:

tensor transformation law

The following equation :

Maxwell equations first group

gives the first group of Maxwell’s equations .


If the Levi-Civita tensor ( e^{\text{ikmn}} ) is used ( a completely antisymmetric 4th rank tensor) , one gets for the first group of equations:

Levi-Civita tensor

Using the expression of the action with the Lagrangian density and taking the variation of the action as stationary:


The last relation above represents the Lagrange equations. And the electromagnetic tensor  Fik is assumed to have been found and defined after developing the Lorentz force.

The relations above give:

Lagrangian equations
Thus Lagrange’s equations give the result:

Lagrange's equations Maxwell  result

And the second group of Maxwell’s equations is given by :

Maxwell equations second group

The second group of equations also gives the equation of charge conservation :

charge conservation
Another notation (the comma notation) and an additional way to write Maxwell’s equations in tensor form are the following :

Maxwell equations tensors comma notation


In a gravitational field the electromagnetic tensor is given by:

electromagnetic tensor gravitation

\nabla _j is the covariant derivative and \Gamma _{\text{ij}}^k  are Christoffel’s symbols .
So the relation between the electromagnetic tensor and the potentials is unchanged in the presence of a gravitational field.

The first and second groups of Maxwell’s equations become :

first and second group Maxwell equations


After calculation the first group can be shown to be equal to the original first group of equations without gravitation.

The second group can be expressed as:

second group with gravitation

where g is the determinant of the metric tensor.

The equations of motion of a particle of charge q in the presence of an electromagnetic field and a gravitational field are given by  :

equations of motion

Or equivalently:

motion of partilce

In General Relativity , the energy-momentum tensor Tij of an electromagnetic field in free space is expressed in the form of the electromagnetic stress- energy tensor:

energy momentum tensor
And the Einstein field equations

Einstein field equations.

are called the Einstein-Maxwell equations.

To be continued…

Question: what is the square root of 36 ? – Part Two

I will continue with answers and results equal to the square root of 36 ( originally answered at ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless .
If the result is a simple fraction with numerator and denominator , then the units usually cancel out.
In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result.
In one or two results where I didn’t look for the inverse units  I multiplied the equation with a quantity I called (U) ,  which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

square root of 36 physics


One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on  the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant ,  unit or number , then they would have likely  tried to construct a system of measurement  based on the number 6 , and  to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 or \sqrt{36} is equal to :

  • The floor of  2 \pi :
    2 \pi \approx 6.2831853071795864769 ;6=\lfloor 2 \pi \rfloor
  • \frac{1}{60} of the circumference of a circle in degrees.
  • It is also  one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of  12 months which make up a year , etc.
  • A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6 (or \sqrt{36} ) .
Or whatever.

Apologies to Isaac Newton , Leonhard Euler , Bernhard Riemann , Einstein , Stokes , Coulomb , Avogadro , Lagrange , and others (wherever they may be) , for playing around with their equations , formulas , constants , and/or functions.

And one more addiction to this answer :

Does the future of humanity depend on answering what is the square root of  36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era  in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are ( 3=\frac{\sqrt{36}}{2} ) more answers to \sqrt{36} , this time with images :

\sqrt{36}  is equal to :

The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

number 6 rotated

The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

number 6 polar plot

The number expressing the degree of the root  and the power of the variables in the 3D plot below :

sinc number 6

The rotated number  and the polar plotted curve in the first two images  above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

Online sources and reference works related to what I have written in this answer can be found in my pages about Science books problem solvers and philosophy books in this site/blog.

Some other online sources: