Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, both classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences, he remained in France until the end of his life. He was involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, became Senator in 1799.
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals. He extended the method to take into account possible constraints, arriving at the method of Lagrange multipliers. Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations, he proved. His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. In calculus, Lagrange developed a novel approach to interpolation and Taylor series, he studied the three-body problem for the Earth and Moon and the movement of Jupiter's satellites, in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points. But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, presented the so-called mechanical "principles" as simple results of the variational calculus.
Born as Giuseppe Lodovico Lagrangia, Lagrange was of French descent. His paternal great-grandfather was a French army officer who had moved to Turin, the de facto capital of the kingdom of Piedmont-Sardinia at Lagrange's time, married an Italian, his mother was from the countryside of Turin. He was raised as a Roman Catholic, his father, who had charge of the king's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained a good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, Lagrange seems to have accepted this willingly, he studied at the University of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident.
Alone and unaided he threw himself into mathematical studies. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange proved to be a problematic professor with his oblivious teaching style, abstract reasoning, impatience with artillery and fortification-engineering applications. In this Academy one of his students was François Daviet de Foncenex. Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Lagrange wrote several letters to Leonhard Euler between 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and simplifying Euler's earlier analysis. Lagrange applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was impressed with Lagrange's results, it has been stated that "with characteristic courtesy he withheld a paper he had written, which covered some of the same ground, in order that the young Italian might have time to complete his work, claim the undisputed invention of the new calculus". Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773. In 1758, with the aid of his pupils, Lagrange established a society, subsequently incorporated as the Turin Aca
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is quantified numerically using the SI derived unit, the cubic metre; the volume of a container is understood to be the capacity of the container. Three dimensional mathematical shapes are assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, circular shapes can be calculated using arithmetic formulas. Volumes of complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space; the volume of a solid can be determined by fluid displacement. Displacement of liquid can be used to determine the volume of a gas; the combined volume of two substances is greater than the volume of just one of the substances. However, sometimes one substance dissolves in the other and in such cases the combined volume is not additive.
In differential geometry, volume is expressed by means of the volume form, is an important global Riemannian invariant. In thermodynamics, volume is a fundamental parameter, is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube. In the International System of Units, the standard unit of volume is the cubic metre; the metric system includes the litre as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = 3 = 1000 cubic centimetres = 0.001 cubic metres. Small amounts of liquid are measured in millilitres, where 1 millilitre = 0.001 litres = 1 cubic centimetre. In the same way, large amounts can be measured in megalitres, where 1 million litres = 1000 cubic metres = 1 megalitre. Various other traditional units of volume are in use, including the cubic inch, the cubic foot, the cubic yard, the cubic mile, the teaspoon, the tablespoon, the fluid ounce, the fluid dram, the gill, the pint, the quart, the gallon, the minim, the barrel, the cord, the peck, the bushel, the hogshead, the acre-foot and the board foot.
Capacity is defined by the Oxford English Dictionary as "the measure applied to the content of a vessel, to liquids, grain, or the like, which take the shape of that which holds them". Capacity is not identical in meaning to volume, though related. Units of capacity are the SI litre and its derived units, Imperial units such as gill, pint and others. Units of volume are the cubes of units of length. In SI the units of volume and capacity are related: one litre is 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial; the density of an object is defined as the ratio of the mass to the volume. The inverse of density is specific volume, defined as volume divided by mass. Specific volume is a concept important in thermodynamics where the volume of a working fluid is an important parameter of a system being studied; the volumetric flow rate in fluid dynamics is the volume of fluid which passes through a given surface per unit time. In calculus, a branch of mathematics, the volume of a region D in R3 is given by a triple integral of the constant function f = 1 and is written as: ∭ D 1 d x d y d z.
The volume integral in cylindrical coordinates is ∭ D r d r d θ d z, the volume integral in spherical coordinates has the form ∭ D ρ 2 sin ϕ d ρ d θ d ϕ. The above formulas can be used to show that the volumes of a cone and cylinder of the same radius and height are in the ratio 1: 2: 3, as follows. Let the radius be r and the height be h the volume of cone is 1 3 π r 2 h = 1 3 π r 2 = × 1, the volume of the sphere
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
A mnemonic device, or memory device, is any learning technique that aids information retention or retrieval in the human memory. Mnemonics make use of elaborative encoding, retrieval cues, imagery as specific tools to encode any given information in a way that allows for efficient storage and retrieval. Mnemonics aid original information in becoming associated with something more accessible or meaningful—which, in turn, provides better retention of the information. Encountered mnemonics are used for lists and in auditory form, such as short poems, acronyms, or memorable phrases, but mnemonics can be used for other types of information and in visual or kinesthetic forms, their use is based on the observation that the human mind more remembers spatial, surprising, sexual, humorous, or otherwise "relatable" information, rather than more abstract or impersonal forms of information. The word "mnemonic" is derived from the Ancient Greek word μνημονικός, meaning "of memory, or relating to memory" and is related to Mnemosyne, the name of the goddess of memory in Greek mythology.
Both of these words are derived from μνήμη, "remembrance, memory". Mnemonics in antiquity were most considered in the context of what is today known as the art of memory. Ancient Greeks and Romans distinguished between two types of memory: the "natural" memory and the "artificial" memory; the former is inborn, is the one that everyone uses instinctively. The latter in contrast has to be trained and developed through the learning and practice of a variety of mnemonic techniques. Mnemonic systems are strategies consciously used to improve memory, they help use information stored in long-term memory to make memorisation an easier task. The general name of mnemonics, or memoria technica, was the name applied to devices for aiding the memory, to enable the mind to reproduce a unfamiliar idea, a series of dissociated ideas, by connecting it, or them, in some artificial whole, the parts of which are mutually suggestive. Mnemonic devices were much cultivated by Greek sophists and philosophers and are referred to by Plato and Aristotle.
In times the poet Simonides was credited for development of these techniques for no reason other than that the power of his memory was famous. Cicero, who attaches considerable importance to the art, but more to the principle of order as the best help to memory, speaks of Carneades of Athens and Metrodorus of Scepsis as distinguished examples of people who used well-ordered images to aid the memory; the Romans valued. The Greek and the Roman system of mnemonics was founded on the use of mental places and signs or pictures, known as "topical" mnemonics; the most usual method was to choose a large house, of which the apartments, windows, furniture, etc. were each associated with certain names, events or ideas, by means of symbolic pictures. To recall these, an individual had only to search over the apartments of the house until discovering the places where images had been placed by the imagination. In accordance with said system, if it were desired to fix a historic date in memory, it was localised in an imaginary town divided into a certain number of districts, each of with ten houses, each house with ten rooms, each room with a hundred quadrates or memory-places on the floor on the four walls on the roof.
Therefore, if it were desired to fix in the memory the date of the invention of printing, an imaginary book, or some other symbol of printing, would be placed in the thirty-sixth quadrate or memory-place of the fourth room of the first house of the historic district of the town. Except that the rules of mnemonics are referred to by Martianus Capella, nothing further is known regarding the practice until the 13th century. Among the voluminous writings of Roger Bacon is a tractate De arte memorativa. Ramon Llull devoted special attention to mnemonics in connection with his ars generalis; the first important modification of the method of the Romans was that invented by the German poet Konrad Celtes, who, in his Epitoma in utramque Ciceronis rhetoricam cum arte memorativa nova, used letters of the alphabet for associations, rather than places. About the end of the 15th century, Petrus de Ravenna provoked such astonishment in Italy by his mnemonic feats that he was believed by many to be a necromancer.
His Phoenix artis memoriae went through as many as nine editions, the seventh being published at Cologne in 1608. About the end of the 16th century, Lambert Schenkel, who taught mnemonics in France and Germany surprised people with his memory, he was denounced as a sorcerer by the University of Louvain, but in 1593 he published his tractate De memoria at Douai with the sanction of that celebrated theological faculty. The most complete account of his system is given in two works by his pupil Martin Sommer, published in Venice in 1619. In 1618 John Willis published Mnemonica. Giordano Bruno included a memoria technica in his treatise De umbris idearum, as part of his study of the ars generalis of Llull. Other writers of this period are the Florentine Publicius. Porta, Ars reminiscendi. In 1648 Stanislaus Mink von Wennsshein revealed what he called the "most fertile secret" in mnemonics — using consonants for figures, thus expressing numbers by words, i
Multiplication is one of the four elementary mathematical operations of arithmetic. The multiplication of whole numbers may be thought as a repeated addition; the multiplier can be multiplicand second. A × b = b + ⋯ + b ⏟ a For example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together: 3 × 4 = 4 + 4 + 4 = 12 Here 3 and 4 are the factors and 12 is the product. One of the main properties of multiplication is the commutative property: adding 3 copies of 4 gives the same result as adding 4 copies of 3: 4 × 3 = 3 + 3 + 3 + 3 = 12 Thus the designation of multiplier and multiplicand does not affect the result of the multiplication; the multiplication of integers, rational numbers and real numbers is defined by a systematic generalization of this basic definition. Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths; the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property.
The product of two measurements is a new type of measurement, for instance multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is defined for other types of numbers, such as complex numbers, more abstract constructs, like matrices. For some of these more abstract constructs, the order in which the operands are multiplied together matters. A listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is written using the sign "×" between the terms. For example, 2 × 3 = 6 3 × 4 = 12 2 × 3 × 5 = 6 × 5 = 30 2 × 2 × 2 × 2 × 2 = 32 The sign is encoded in Unicode at U+00D7 × MULTIPLICATION SIGN. There are other mathematical notations for multiplication: Multiplication is denoted by dot signs a middle-position dot:5 ⋅ 2 or 5.
3 The middle dot notation, encoded in Unicode as U+22C5 ⋅ DOT OPERATOR, is standard in the United States, the United Kingdom, other countries where the period is used as a decimal point. When the dot operator character is not accessible, the interpunct is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication. In algebra, multiplication involving variables is written as a juxtaposition called implied multiplication; the notation can be used for quantities that are surrounded by parentheses. This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations. In vector multiplication, there is a distinction between the dot symbols; the cross symbol denotes the taking a cross product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar.
In computer programming, the asterisk is still the most common notation. This is due to the fact that most computers were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard; this usage originated in the FORTRAN programming language. The numbers to be multiplied are called the "factors"; the number to be multiplied is the "multiplicand", the number by which it is multiplied is the "multiplier". The multiplier is placed first and the multiplicand is placed second; as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a elementary level and
Kiyosi Itô was a Japanese mathematician. He pioneered the theory of stochastic integration and stochastic differential equations, now known as the Itô calculus, its basic concept is the Itô integral, among the most important results is a change of variable formula known as Itô's lemma. Itô calculus is a method used in the mathematical study of random events and is applied in various fields, is best known for its use in mathematical finance. Ito made contributions to the study of diffusion processes on manifolds, known as stochastic differential geometry. Although the standard Hepburn romanization of his name is Kiyoshi Itō, he used the spelling Kiyosi Itô; the alternative spellings Itoh and Ito are sometimes seen in the West. Itô was born in Hokusei in Mie Prefecture on the main island of Honshū, he graduated with a B. S. and a Ph. D in Mathematics from the University of Tokyo. Between 1938 and 1945, Itô worked for the Japanese National Statistical Bureau, where he published two of his seminal works on probability and stochastic processes.
After that he continued to develop his ideas on stochastic analysis with many important papers on the topic. In 1952, he became a Professor at the University of Kyoto where he remained until his retirement in 1979. Starting in the 1950s, Itô spent long periods of time outside Japan at Cornell, the Institute for Advanced Study in Princeton, N. J. and Aarhus University in Denmark. Itô was awarded the inaugural Gauss Prize in 2006 by the International Mathematical Union for his lifetime achievements; as he was unable to travel to Madrid, his youngest daughter, Junko Itô received the Gauss Prize from the King of Spain on his behalf. International Mathematics Union President Sir John Ball presented the medal to Itô at a special ceremony held in Kyoto. In October 2008, Itô was honored with Japan's Order of Culture. Itô wrote in Japanese, German and English. Itô died on November 10, 2008 in Kyoto, Japan at age 93. Itô diffusion Itô isometry Black–Scholes model Kiyosi Ito. "On the Probability Distribution on a Compact Group".
Nippon Sugaku-Buturigakkwai Kizi Dai 3 Ki / Proceedings of the Physico-Mathematical Society of Japan. 3rd series. 22: 977–998. Kiyosi Ito. "Differential equations determining a Markoff process". Zenkoku Sizyo Sugaku Danwakai-si: 1352–1400. Kiyosi Ito. "Stochastic integral". Proceedings of the Imperial Academy. 20: 519–524. Doi:10.3792/pia/1195572786. Kiyosi Ito. "On a stochastic integral equation". Proceedings of the Japan Academy. 22: 32–35. Doi:10.3792/pja/1195572371. Kiyosi Ito. "Stochastic differential equations in a differentiable manifold". Nagoya Mathematical Journal. 1: 35–47. Kiyosi Ito. "On a formula concerning stochastic differentials". Nagoya Mathematical Journal. 3: 55–65. Kiyosi Ito and Henry McKean. Diffusion Processes and Their Sample Paths. Berlin: Springer Verlag. ISBN 978-3-540-60629-1. Kiyosi Ito. Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-193-6. Bibliography of Kiyosi Itô Kiyosi Itô at Research Institute for Mathematical Sciences Kiyosi Itô at the Mathematics Genealogy Project
Orientation (vector space)
In mathematics, orientation is a geometric notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary finite dimension. In this setting, the orientation of an ordered basis is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple rotation. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a rotation alone, but it is possible to do so by reflecting the figure in a mirror; as a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed. The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are declared to be positively oriented, but the choice is arbitrary, as they may be assigned a negative orientation.
A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. Let V be a finite-dimensional real vector space and let b1 and b2 be two ordered bases for V, it is a standard result in linear algebra that there exists a unique linear transformation A: V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation; the property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are two equivalence classes determined by this relation. An orientation on V is an assignment of − 1 to the other; every ordered basis lives in another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on Rn provides a standard orientation on Rn. Any choice of a linear isomorphism between V and Rn will provide an orientation on V.
The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some permutation, they will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Let A be a nonsingular linear mapping of vector space Rn to Rn; this mapping is orientation-preserving. For instance, in R3 a rotation around the Z Cartesian axis by an angle α is orientation-preserving: A 1 = while a reflection by the XY Cartesian plane is not orientation-preserving: A 2 = The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has the zero vector; the only basis of a zero-dimensional vector space is the empty set ∅. Therefore, there is a single equivalence class of ordered bases, the class whose sole member is the empty set; this means that an orientation of a zero-dimensional space is a function →. It is therefore possible to orient a point in two different ways and negative.
Because there is only a single ordered basis ∅, a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing ↦ + 1 or ↦ − 1 therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they preserve the orientation; this is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the fundamental theorem of calculus as an instance of Stokes' theorem. A closed interval is a one-dimensional m