Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.
For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.
Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.
He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.
Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer, it states that for any continuous function f mapping a compact convex set to itself there is a point x 0 such that f = x 0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself. Among hundreds of fixed-point theorems, Brouwer's is well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem; this gives it a place among the fundamental theorems of topology. The theorem is used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.
It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu; the theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods; this work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Luitzen Egbertus Jan Brouwer; the theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: In the plane Every continuous function from a closed disk to itself has at least one fixed point; this can be generalized to an arbitrary finite dimension: In Euclidean space Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
A more general version is as follows: Convex compact set Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point. An more general form is better known under a different name: Schauder fixed point theorem Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point; the theorem holds only for sets that are convex. The following examples show. Consider the function f = x + 1, a continuous function from R to itself; as it shifts every point to the right, it cannot have a fixed point. Note that R is convex and closed, but not bounded. Consider the function f = x + 1 2, a continuous function from the open interval to itself. In this interval, it shifts every point to the right, so it cannot have a fixed point. Note, convex and bounded, but not closed; the function f does have a fixed point for the closed interval, namely f = 1. Note that convexity is not necessary for BFPT; because the properties involved are invariant under homeomorphisms, BFPT is equivalent to forms in which the domain is required to be a closed unit ball D n.
For the same reason it holds for every set, homeomorphic to a closed ball. The following example shows. Consider the following function, defined in polar coordinates: f =, a continuous function from the unit circle to itself, it rotates every point on the unit circle 45 degrees counterclockwise, so it cannot have a fixed point. Note that the unit circle is closed and bounded, but it has a hole; the function f does have a fixed point for the unit disc. A formal generalization of BFPT for "hole-free" domains can be derived from the Lefschetz fixed-point theorem; the continuous function in this theorem is not required to be bijective or surjective. The theorem has several "real world" illustrations. Here are some examples. 1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will be at least one point of the crumpled sheet that lies directly above its corresponding point of the flat sheet.
This is a consequence of the n = 2 case o
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis, he was the founder of the mathematical philosophy of intuitionism. Early in his career, Brouwer proved a number of theorems in the emerging field of topology; the most important were his fixed point theorem, the topological invariance of degree, the topological invariance of dimension. Among mathematicians the best known is the first one referred to now as the Brouwer Fixed Point Theorem, it is a simple corollary to the second, concerning the topological invariance of degree, the best known among algebraic topologists. The third theorem is the hardest. Brouwer proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.
In 1912, at age 31, he was elected a member of the Royal Netherlands Academy of Sciences. He was an Invited Speaker of the ICM in 1908 at Rome and in 1912 at Cambridge, UK. Brouwer founded intuitionism, a philosophy of mathematics that challenged the then-prevailing formalism of David Hilbert and his collaborators, who included Paul Bernays, Wilhelm Ackermann, John von Neumann. A variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics, it is sometimes and rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning. Brouwer was a member of the Significs Group, it formed part of the early history of semiotics—the study of symbols—around Victoria, Lady Welby in particular. The original meaning of his intuitionism can not be disentangled from the intellectual milieu of that group. In 1905, at the age of 24, Brouwer expressed his philosophy of life in a short tract Life and Mysticism, described by the mathematician Martin Davis as "drenched in romantic pessimism".
Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Brouwer "embarked on a self-righteous campaign to reconstruct mathematical practice from the ground up so as to satisfy his philosophical convictions". In 1908: "... Brouwer, in a paper entitled'The untrustworthiness of the principles of logic', challenged the belief that the rules of the classical logic, which have come down to us from Aristotle have an absolute validity, independent of the subject matter to which they are applied"."After completing his dissertation, Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess". Hilbert—the formalist with whom the intuitionist Brouwer would spend years in conflict—admired the young man and helped him receive a regular academic appointment at the University of Amsterdam, it was that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism ".
He was combative as a young man. He was involved in a public and demeaning controversy in the 1920s with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal, he became isolated. Dutch mathematician and historian of mathematics, Bartel Leendert van der Waerden attended lectures given by Brouwer in years, commented: "Even though his most important research contributions were in topology, Brouwer never gave courses in topology, but always on—and only on—the foundations of his intuitionism, it seemed that he was no longer convinced of his results in topology because they were not correct from the point of view of intuitionism, he judged everything he had done before, his greatest output, false according to his philosophy."About his last years, Davis remarks: "...he felt more and more isolated, spent his last years under the spell of'totally unfounded financial worries and a paranoid fear of bankruptcy and illness.' He was killed in 1966 at the age of 85, struck by a vehicle while crossing the street in front of his house."
Jean van Heijenoort, 1967 3rd printing 1976 with corrections, A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 pbk; the original papers are prefaced with valuable commentary. 1923. L. E. J. Brouwer: "On the significance of the principle of excluded middle in mathematics in function theory." With two Addenda and corrigenda, 334-45. Brouwer gives brief synopsis of his belief that the law of excluded middle cannot be "applied without reservation in the mathematics of infinite systems" and gives two examples of failures to illustrate his assertion. 1925. A. N. Kolmogorov: "On the principle of excluded middle", pp. 414–437. Kolmogorov disputes a few.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane.
When a region appears in two neighbouring charts, the two representations do not coincide and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t
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