In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature; the term "minimal surface" is used because these surfaces arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may exist several minimal surfaces with different areas: the standard definitions only relate to a local optimum, not a global optimum. Minimal surfaces can be defined in several equivalent ways in R3; the fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.
Local least area definition: A surface M ⊂ R3 is minimal if and only if every point p ∈ M has a neighbourhood with least-area relative to its boundary. Note that this property is local: there might exist other surfaces that minimize area better with the same global boundary. Variational definition: A surface M ⊂ R3 is minimal if and only if it is a critical point of the area functional for all compactly supported variations; this definition makes minimal surfaces a 2-dimensional analogue to geodesics. Soap film definition: A surface M ⊂ R3 is minimal if and only if every point p ∈ M has a neighbourhood Dp, equal to the unique idealized soap film with boundary ∂DpBy the Young–Laplace equation the curvature of a soap film is proportional to the difference in pressure between the sides: if it is zero, the membrane has zero mean curvature. Note that spherical bubbles are not minimal surfaces as per this definition: while they minimize total area subject to a constraint on internal volume, they have a positive pressure.
Mean curvature definition: A surface M ⊂ R3 is minimal if and only if its mean curvature vanishes identically. A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. Differential equation definition: A surface M ⊂ R3 is minimal if and only if it can be locally expressed as the graph of a solution of u y y − 2 u x u y u x y + u x x = 0 The partial differential equation in this definition was found in 1762 by Lagrange, Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature. Energy definition: A conformal immersion X: M → R3 is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point p ∈ M has a neighbourhood with least energy relative to its boundary; this definition ties minimal surfaces to potential theory. Harmonic definition: If X =: M → R3 is an isometric immersion of a Riemann surface into 3-space X is said to be minimal whenever xi is a harmonic function on M for each i.
A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in R3. Gauss map definition: A surface M ⊂ R3 is minimal if and only if its stereographically projected Gauss map g: M → C ∪ is meromorphic with respect to the underlying Riemann surface structure, M is not a piece of a sphere; this definition uses that the mean curvature is half of the trace of the shape operator, linked to the derivatives of the Gauss map. If the projected Gauss map obeys the Cauchy–Riemann equations either the trace vanishes or every point of M is umbilic, in which case it is a piece of a sphere. Mean curvature flow definition: Minimal surfaces are the critical points for the mean curvature flow; the local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R3. Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z of least area stretched across a given closed contour.
He derived the Euler–Lagrange equation for the solution d d x + d d y = 0 He did not succeed in finding any solution beyond the plane. In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-min
Weierstrass's elliptic functions
In mathematics, Weierstrass's elliptic functions are elliptic functions that take a simple form. This class of functions are referred to as p-functions and written using the symbol ℘; the ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori. Genus one solutions of differential equations can be written in terms of Weierstrass elliptic functions. Notably, the simplest periodic solutions of the Korteweg–de Vries equation are written in terms of Weierstrass p-functions; the Weierstrass elliptic function can be defined in three related ways, each of which possesses certain advantages. One is as a lattice Λ in the complex plane. Another is in terms of z and two complex numbers ω1 and ω2 defining a pair of generators, or periods, for the lattice; the third is in a modulus τ in the upper half-plane. This is related to the previous definition by τ = ω2/ω1, which by the conventional choice on the pair of periods is in the upper half-plane.
Using this approach, for fixed z the Weierstrass functions become modular functions of τ. In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods ω1 and ω2 defined as ℘ = 1 z 2 + ∑ n 2 + m 2 ≠ 0. Λ = are the points of the period lattice, so that ℘ = ℘ for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice. If τ is a complex number in the upper half-plane ℘ = ℘ = 1 z 2 + ∑ n 2 + m 2 ≠ 0; the above sum is homogeneous of degree minus two, from which we may define the Weierstrass ℘ function for any pair of periods, as ℘ = ℘ ω 1 2. We may compute ℘ rapidly in terms of theta functions; the formula here is ℘ = π 2 ϑ 2 ϑ 10 2 ϑ 01 2 ϑ 11 2 − π 2 3 [ ϑ 4 + ϑ 10 4 (
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function, holomorphic on all of D except for a discrete set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros, meaning "part," as opposed to holos, meaning "whole." Every meromorphic function on D can be expressed as the ratio between two holomorphic functions defined on D: any pole must coincide with a zero of the denominator. Intuitively, a meromorphic function is a ratio of two well-behaved functions; such a function will still be well-behaved, except at the points where the denominator of the fraction is zero. If the denominator has a zero at z and the numerator does not the value of the function will approach infinity. From an algebraic point of view, if D is connected the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions; this is analogous to the relationship between the integers.
In the 1930s, in group theory, a meromorphic function was a function from a group G into itself that preserved the product on the group. The image of this function was called an automorphism of G. Similarly, a homomorphic function was a function between groups that preserved the product, while a homomorphism was the image of a homomorph; this terminology is now obsolete. The term endomorphism is now used for the function itself, with no special name given to the image of the function; the term meromorph is no longer used in group theory. Since the poles of a meromorphic function are isolated, there are at most countably many; the set of poles can be infinite, as exemplified by the function f = csc z = 1 sin z. By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted and the quotient f / g can be formed unless g = 0 on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers.
In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f = z 1 / z 2 is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two. Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori. All rational functions, for example f = z 3 − 2 z + 10 z 5 + 3 z − 1, are meromorphic on the whole complex plane; the functions f = e z z and f = sin z 2 as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. The function f = e 1 z is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane.
However, it is meromorphic on C ∖. The complex logarithm function f = ln is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points; the function f = csc 1 z = 1 sin is not meromorphic in the whole plane, since the point z = 0 is an accumulation point of poles and is thus not an isolated singularity. The function f = sin 1
Karl Theodor Wilhelm Weierstrass was a German mathematician cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a teacher teaching mathematics, physics and gymnastics. Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born in part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, Theodora Vonderforst, his interest in mathematics began. He was sent to the University of Bonn upon graduation to prepare for a government position; because his studies were to be in the fields of law and finance, he was in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics.
The outcome was to leave the university without a degree. After that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster, he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he taught physics and gymnastics. Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a long period of illness, but was able to publish papers that brought him fame and distinction; the University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin.
At the age of fifty-five, Weierstrass met Sonya Kovalevsky whom he tutored after failing to secure her admission to the University. They had a fruitful intellectual, but troubled personal relationship that "far transcended the usual teacher-student relationship"; the misinterpretation of this relationship and Kovalevsky's early death in 1891 was said to have contributed to Weierstrass' ill-health. He was immobile for the last three years of his life, died in Berlin from pneumonia. Weierstrass was interested in the soundness of calculus, at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 his work remained unknown to most of the mathematical community until years and many mathematicians had only vague definitions of limits and continuity of functions. Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.
Cauchy did not distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars; the correct statement is. This required the concept of uniform convergence, first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, both formalized it and applied it throughout the foundations of calculus; the formal definition of continuity of a function, as formulated by Weierstrass, is as follows: f is continuous at x = x 0 if ∀ ε > 0 ∃ δ > 0 such that for every x in the domain of f, | x − x 0 | < δ ⇒ | f − f | < ε. In simple English, f is continuous at a point x = x 0 if for each ε > 0 there exists a δ > 0 such that the function f lies between f − ε and f
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to