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Helmholtz | A Wisdom Archive on Helmholtz | | Helmholtz A selection of articles related to Helmholtz | |
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| ARTICLES RELATED TO Helmholtz | | | |  Helmholtz: Encyclopedia II - History of calculus - Controversy Newton Leibnitz... or Madhava?Madhava of Sangamagrama and the Kerala school were the first to come up with the important ideas of calculus in the 14th century and some [3] propose these ideas may have been transmitted to Europe by the 17th century. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.
In the controversy between Newton and Leibniz, suggestions were made that the work of Leibniz was not independent, as he claimed, but in ...
See also:History of calculus, History of calculus - Invention of Calculus, History of calculus - Controversy Newton Leibnitz... or Madhava?, History of calculus - Rigorous foundations, History of calculus - Integrals, History of calculus - Symbolic methods, History of calculus - Calculus of variations, History of calculus - Applications Read more here: » History of calculus: Encyclopedia II - History of calculus - Controversy Newton Leibnitz... or Madhava? |
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| | | | | Helmholtz: Encyclopedia II - Germany - Politics
Germany - Legal system.
Germany has a civil or statute law system based ultimately on Roman law. Legislative power is divided between the Federation and the individual federated states. While criminal law and private law have seen codifications on the national level (in the Strafgesetzbuch and the Bürgerliches Gesetzbuch respectively), no such unifying codification exists in administrative law where a lot of the fundamental matters remain in the jurisdiction of the individual federated states. Ther ...
See also:Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire of the German Nation 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - States Länder, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics Read more here: » Germany: Encyclopedia II - Germany - Politics |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Applications An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics
HΨE = EΨE
where H, the Hamiltonian, is a second-order differential operator and ΨE, the ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Applications |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Definitions Transformations of space—such as translation (or shifting the origin), rotation, reflection, stretching, compression, or any combination of these; other transformations could also be listed—may be visualized by the effect they produce on vectors. Vectors can be visualised as arrows pointing from one point to another.
Eigenvectors of transformations are vectors[3] which are either left unaffected or simply multiplied by a scale fa ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Definitions |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Examples As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation: An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the equator would not be an eigenvector. Since the arrow pointing at the pole i ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Examples |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces If the vector space is infinite dimensional, it may be advantageous to define the concept of spectral values. The spectral values are the set of scalars λ for which the Green's operator, , associated to the transformation is not defined, that is such that is not invertible (i.e., the inverse transformation to does not exist).
If λ is an eigenvalue of , λ is also a spectral value of it. However, the reverse relation is not true: any spectral value is not an eigenvalue. There are operators on Hilbert or Banach ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Spectral theorem The spectral theorem depicts the whole importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions, a linear transformation of a vector can be expressed as the linear combination of the eigenvectors with coefficients equal to the eigenvalues times the scalar prod ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Spectral theorem |
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| | | | | Helmholtz: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Eigenvalue equation Mathematically, vλ is an eigenvector and λ the corresponding eigenvalue of a transformation if the equation:
is true, where is the vector obtained when applying the transformation to vλ.
Suppose is a linear transformation (which means that for all scalars a, b, and vectors v, w). Consider a basis in that vector space. Then, and vλ can be represented relative to that basis by a matrix T< ...
See also:Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Eigenvalue equation |
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| | | | | Helmholtz: Encyclopedia II - Legendre transformation - Examples The exponential function ex has x ln x − x as a Legendre transform since the respective first derivatives ex and ln x are inverse to each other. This example shows that the respective domains of a function and its Legendre transform need not agree.
Similarly, the quadratic form
with A a symmetric invertible n-by-n-matrix has< ...
See also:Legendre transformation, Legendre transformation - Applications, Legendre transformation - Examples, Legendre transformation - Legendre transformation in one dimension, Legendre transformation - Geometric interpretation, Legendre transformation - Legendre transformation in more than one dimension, Legendre transformation - Further properties, Legendre transformation - Scaling properties, Legendre transformation - Behavior under translation, Legendre transformation - Behavior under inversion, Legendre transformation - Behavior under linear transformations, Legendre transformation - Infimal convolution Read more here: » Legendre transformation: Encyclopedia II - Legendre transformation - Examples |
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| | | | | Helmholtz: Encyclopedia II - Legendre transformation - Legendre transformation in one dimension In one dimension, a Legendre transform to a function f : R → R with an invertible first derivative may be found using the formula
This can be seen by integrating both sides of the defining condition restricted to one-dimension
from x0 to x1, making use of the fundamental theorem of calculus on the left hand side and substitu ...
See also:Legendre transformation, Legendre transformation - Applications, Legendre transformation - Examples, Legendre transformation - Legendre transformation in one dimension, Legendre transformation - Geometric interpretation, Legendre transformation - Legendre transformation in more than one dimension, Legendre transformation - Further properties, Legendre transformation - Scaling properties, Legendre transformation - Behavior under translation, Legendre transformation - Behavior under inversion, Legendre transformation - Behavior under linear transformations, Legendre transformation - Infimal convolution Read more here: » Legendre transformation: Encyclopedia II - Legendre transformation - Legendre transformation in one dimension |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Einstein and the Theory of Relativity In 1905 the three epochal papers of hitherto completely unknown Albert Einstein
were published in the journal Annalen der Physik; Planck was among the few who immediately recognized the significance of the special theory of relativity. Thanks to his influence this theory was soon widely accepted in Germany. Planck also contributed considerably to extend the special theory of relativity.
However, Einstein's hypothesis of light quanta (photons), harbingered by Philipp Lenard's 1902 discovery of the photoelectric effect, was initially r ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Einstein and the Theory of Relativity |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Black-body radiation In 1894 Planck turned his attention to the problem of black-body radiation. He had been commissioned by electric companies to discover how to create the most light from lightbulbs with the minimum energy. The problem had already been stated by Kirchhoff in 1859: How does the intensity of the electromagnetic radiation emitted by a black body (a perfect absorber, also known as a cavity radiator) depend on the frequency of the radiation (e.g., the colour of the light) and the temperature of the body? The question had been explored experimenta ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Black-body radiation |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Family In March 1887 Planck married Marie Merck (1861-1909), sister of a school fellow, and
moved with her into a sublet apartment in Kiel. Four children were born to the couple:
Karl (1888-1916), the twins Emma (1889-1919) and Grete (1889-1917), and Erwin (1893-1945).
After the appointment to Berlin the Planck family lived in a villa in Berlin-Grunewald, Wangenheimstraße 21. In the vicinity of this address several other professors of Berlin University
were living, among them the famous theologian Adolf von Harnack, who became a close frie ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Family |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Academic career With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he would be offered an academic position. Although he was initially ignored by the academic community, he furthered his work on the field of heat theory and discovered one after the other the same thermodynamical formalism as Gibbs without realizing it. Clausius's ideas on entropy occupied a central role in his work.
In April 1885 the University of Kiel appointed Planck an associate professor of theoretical physics. Fu ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Academic career |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Nazi dictatorship and Second World War When the Nazis seized power in 1933, Planck had already reached the age of 74; he had to witness how many Jewish friends and colleagues were expelled from their positions and humiliated, and how hundreds of scientists emigrated from Germany. Again he tried "persevere and continue working"
and asked scientists who were considering emigration to stay in Germany. In some cases he was successful with this request, such as in the case of Heisenberg.
Hahn asked Planck whether they should gather a number of well-known German professo ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Nazi dictatorship and Second World War |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Final years After the war, a number of German physicists assembled in Göttingen in order to reestablish the
Kaiser-Wilhelm-Gesellschaft. In July of 1945, Planck agreed to act formally as its president, again. The British occupation authorities insisted on changing the name, and therefore in February 1948 the Max-Planck-Gesellschaft was established.
Despite his deteriorating health, Planck resumed travelling in order to give public talks. In 1946, he went to London on the occasion of the 300th birthday of Isaac Newton. He was the only German invited.
On April 1st of 1946, Planck was succe ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Final years |
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| | | | | Helmholtz: Encyclopedia II - Musical acoustics - Harmonics and non-linearities When a periodic wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), the summed wave is symmetrical; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (0f, 2f, 4f, 6f, ...), it will be asymmetrical; the top half will not be a mirror image of the bottom.
The opposite is also true. A system which changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics. This is called a non-linear system. If it affects the wave symmetrically, the harmonics produced will only be odd, if asymmetrically, a ...
See also:Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics and non-linearities |
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| | | | | Helmholtz: Encyclopedia II - Musical acoustics - Harmonics partials and overtones The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials.
Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere ...
See also:Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics partials and overtones |
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| | | | | Helmholtz: Encyclopedia II - Musical acoustics - Sound waves Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called "sound". Most sound that people recognize as "musical" is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease ...
See also:Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Sound waves |
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| | | | | Helmholtz: Encyclopedia II - Max Planck - Origin and youth Planck came from a traditional, intellectual family. His paternal great-grandfather and grandfather were both theology professors in Göttingen, his father was a law professor in Kiel and Munich, and his paternal uncle was a judge.
Max Planck was born in Kiel on April 23, 1858 to Johann Julius Wilhelm Planck and his second wife, Emma Patzig. He was the sixth child in the family, though two of his siblings were from his father's first marriage. In 1867 the family moved to Munich, where Planck at ...
See also:Max Planck, Max Planck - Origin and youth, Max Planck - Education, Max Planck - Academic career, Max Planck - Family, Max Planck - Professor at Berlin University, Max Planck - Black-body radiation, Max Planck - Einstein and the Theory of Relativity, Max Planck - World War and Weimar Republic, Max Planck - Quantum Mechanics, Max Planck - Nazi dictatorship and Second World War, Max Planck - Final years, Max Planck - Honours and medals Read more here: » Max Planck: Encyclopedia II - Max Planck - Origin and youth |
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| | | | | Helmholtz: Encyclopedia II - Legendre transformation - Legendre transformation in more than one dimension For a differentiable real-valued function on an open subset U of Rn the Legendre conjugate of the pair (U, f) is defined to be the pair (V, g), where V is the image of U under the gradient mapping Df, and g is the function on V given by the formula
where
is the scalar product on Rn.
Alternatively, if X is a real vector space and Y is its dual vector space, then ...
See also:Legendre transformation, Legendre transformation - Applications, Legendre transformation - Examples, Legendre transformation - Legendre transformation in one dimension, Legendre transformation - Geometric interpretation, Legendre transformation - Legendre transformation in more than one dimension, Legendre transformation - Further properties, Legendre transformation - Scaling properties, Legendre transformation - Behavior under translation, Legendre transformation - Behavior under inversion, Legendre transformation - Behavior under linear transformations, Legendre transformation - Infimal convolution Read more here: » Legendre transformation: Encyclopedia II - Legendre transformation - Legendre transformation in more than one dimension |
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| | | | | Helmholtz: Encyclopedia II - Legendre transformation - Geometric interpretation For a strictly convex function the Legendre-transformation can be interpreted as a mapping between the graph of the function and the family of tangents of the graph. (The tangents are well-defined at all but at most countably many points since a convex function is differentiable at all but at most countably many points.)
The equation of a line with slope m and y-intercept b is given by
y = mx + b
For this line to be tangent to the graph of a function f at the point (x0< ...
See also:Legendre transformation, Legendre transformation - Applications, Legendre transformation - Examples, Legendre transformation - Legendre transformation in one dimension, Legendre transformation - Geometric interpretation, Legendre transformation - Legendre transformation in more than one dimension, Legendre transformation - Further properties, Legendre transformation - Scaling properties, Legendre transformation - Behavior under translation, Legendre transformation - Behavior under inversion, Legendre transformation - Behavior under linear transformations, Legendre transformation - Infimal convolution Read more here: » Legendre transformation: Encyclopedia II - Legendre transformation - Geometric interpretation |
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