A screw thread shortened to thread, is a helical structure used to convert between rotational and linear movement or force. A screw thread is a ridge wrapped around a cylinder or cone in the form of a helix, with the former being called a straight thread and the latter called a tapered thread. A screw thread is the essential feature of the screw as a simple machine and as a fastener; the mechanical advantage of a screw thread depends on its lead, the linear distance the screw travels in one revolution. In most applications, the lead of a screw thread is chosen so that friction is sufficient to prevent linear motion being converted to rotary, so the screw does not slip when linear force is applied, as long as no external rotational force is present; this characteristic is essential to the vast majority of its uses. The tightening of a fastener's screw thread is comparable to driving a wedge into a gap until it sticks fast through friction and slight elastic deformation. Screw threads have several applications: Fastening: Fasteners such as wood screws, machine screws and bolts.
Connecting threaded hoses to each other and to caps and fixtures. Gear reduction via worm drives Moving objects linearly by converting rotary motion to linear motion, as in the leadscrew of a jack. Measuring by correlating linear motion to rotary motion, as in a micrometer. Both moving objects linearly and measuring the movement, combining the two aforementioned functions, as in a leadscrew of a lathe. In all of these applications, the screw thread has two main functions: It converts rotary motion into linear motion, it prevents linear motion without the corresponding rotation. Every matched pair of threads and internal, can be described as male and female. For example, a screw has male threads; this property is called gender. The helix of a thread can twist in two possible directions, known as handedness. Most threads are oriented so that the threaded item, when seen from a point of view on the axis through the center of the helix, moves away from the viewer when it is turned in a clockwise direction, moves towards the viewer when it is turned counterclockwise.
This is known as a right-handed thread. Threads oriented in the opposite direction are known as left-handed. By common convention, right-handedness is the default handedness for screw threads. Therefore, most threaded parts and fasteners have right-handed threads. Left-handed thread applications include: Where the rotation of a shaft would cause a conventional right-handed nut to loosen rather than to tighten due to applied torque or to fretting induced precession. Examples include: The left hand pedal on a bicycle; the left-hand grinding wheel on a bench grinder. The axle nuts, or less lug nuts on the left side of some automobiles; the securing nut on some circular saw blades – the large torque at startup should tend to tighten the nut. The spindle on brushcutter and line trimmer heads, so that the torque tends to tighten rather than loosen the connection In combination with right-hand threads in turnbuckles and clamping studs. In some gas supply connections to prevent dangerous misconnections, for example: In gas welding the flammable gas supply uses left-handed threads, while the oxygen supply if there is one has a conventional thread The POL valve for LPG cylinders In a situation where neither threaded pipe end can be rotated to tighten or loosen the joint.
In such a case, the coupling will have one left-handed thread. In some instances, for example early ballpoint pens. In mechanisms to give a more intuitive action as: The leadscrew of the cross slide of a lathe to cause the cross slide to move away from the operator when the leadscrew is turned clockwise; the depth of cut screw of a "Bailey" type metal plane for the blade to move in the direction of a regulating right hand finger. Some Edison base lamps and fittings have a left-hand thread to deter theft, since they cannot be used in other light fixtures; the cross-sectional shape of a thread is called its form or threadform. It may be square, trapezoidal, or other shapes; the terms form and threadform sometimes refer to all design aspects taken together. Most triangular threadforms are based on an isosceles triangle; these are called V-threads or vee-threads because of the shape of the letter V. For 60° V-threads, the isosceles triangle is, more equilateral. For buttress threads, the triangle is scalene.
The theoretical triangle is truncated to varying degrees. A V-thread in which there is no truncation is called a sharp V-thread. Truncation occurs for practical reasons—the thread-cutting or thread-forming tool cannot have a sharp point, truncation is desirable anyway, because otherwise: The cutting or forming tool's edge will break too easily; the point of the threadform adds little strength to the thre
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura. In English, Brook Taylor presented his Linear Perspective in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry". In a second book, New Principles of Linear Perspective, Taylor wrote When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure; the Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays.
And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone In projective geometry the points of a line are called a projective range, the set of lines in a plane on a point is called a pencil. Given two lines ℓ and m in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of ℓ and the range of m determined by the lines of the pencil on P is called a perspectivity. A special symbol has been used to show that points Y are related by a perspectivity. In this notation, to show that the center of perspectivity is P, write X ⩞ P Y. Using the language of functions, a central perspectivity with center P is a function f P: ↦ defined by f P = Y whenever P ∈ X Y; this map is an involution, that is, f P = X for all X ∈. The existence of a perspectivity means; the dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range. The composition of two perspectivities is, in general, not a perspectivity.
A perspectivity or a composition of two or more perspectivities is called a projectivity. There are several results concerning projectivities and perspectivities which hold in any pappian projective plane:Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities. Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities. Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity; the bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will be called perspectivities. Let Sm and Tm be two distinct m-dimensional projective spaces contained in an n-dimensional projective space Rn. Let Pn−m−1 be an -dimensional subspace of Rn with no points in common with either Sm or Tm. For each point X of Sm, the space L spanned by X and Pn-m-1 meets Tm in a point Y = fP.
This correspondence fP is called a perspectivity. The central perspectivity described above is the case with n = 2 and m = 1. Let S2 and T2 be two distinct projective planes in a projective 3-space R3. With O and O* being points of R3 in neither plane, use the construction of the last section to project S2 onto T2 by the perspectivity with center O followed by the projection of T2 back onto S2 with the perspectivity with center O*; this composition is a bijective map of the points of S2 onto itself which preserves collinear points and is called a perspective collineation. Let φ be a perspective collineation of S2; each point of the line of intersection of S2 and T2 will be fixed by φ and this line is called the axis of φ. Let point P be the intersection of line OO* with the plane S2. P is fixed by φ and every line of S2 that passes through P is stabilized by φ. P is called the center of φ; the restriction of φ to any line of S2 not passing through P is the central perspectivity in S2 with center P between that line and the line, its image under φ.
Andersen, Brook Taylor's Work on Linear Perspective, Springer, ISBN 0-387-97486-5 Coxeter, Harold Scott MacDonald, Introduction to Geometry, New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930 Fishback, W. T. Projective and Euclidean Geometry, John Wiley & Sons Pedoe, Geometry/A Comprehensive Course, Dover, ISBN 0-486-65812-0 Young, John Wesley, Projective Geometry, The Carus Mathematical M
Biology is the natural science that studies life and living organisms, including their physical structure, chemical processes, molecular interactions, physiological mechanisms and evolution. Despite the complexity of the science, there are certain unifying concepts that consolidate it into a single, coherent field. Biology recognizes the cell as the basic unit of life, genes as the basic unit of heredity, evolution as the engine that propels the creation and extinction of species. Living organisms are open systems that survive by transforming energy and decreasing their local entropy to maintain a stable and vital condition defined as homeostasis. Sub-disciplines of biology are defined by the research methods employed and the kind of system studied: theoretical biology uses mathematical methods to formulate quantitative models while experimental biology performs empirical experiments to test the validity of proposed theories and understand the mechanisms underlying life and how it appeared and evolved from non-living matter about 4 billion years ago through a gradual increase in the complexity of the system.
See branches of biology. The term biology is derived from the Greek word βίος, bios, "life" and the suffix -λογία, -logia, "study of." The Latin-language form of the term first appeared in 1736 when Swedish scientist Carl Linnaeus used biologi in his Bibliotheca botanica. It was used again in 1766 in a work entitled Philosophiae naturalis sive physicae: tomus III, continens geologian, phytologian generalis, by Michael Christoph Hanov, a disciple of Christian Wolff; the first German use, was in a 1771 translation of Linnaeus' work. In 1797, Theodor Georg August Roose used the term in the preface of a book, Grundzüge der Lehre van der Lebenskraft. Karl Friedrich Burdach used the term in 1800 in a more restricted sense of the study of human beings from a morphological and psychological perspective; the term came into its modern usage with the six-volume treatise Biologie, oder Philosophie der lebenden Natur by Gottfried Reinhold Treviranus, who announced: The objects of our research will be the different forms and manifestations of life, the conditions and laws under which these phenomena occur, the causes through which they have been effected.
The science that concerns itself with these objects we will indicate by the name biology or the doctrine of life. Although modern biology is a recent development, sciences related to and included within it have been studied since ancient times. Natural philosophy was studied as early as the ancient civilizations of Mesopotamia, the Indian subcontinent, China. However, the origins of modern biology and its approach to the study of nature are most traced back to ancient Greece. While the formal study of medicine dates back to Hippocrates, it was Aristotle who contributed most extensively to the development of biology. Important are his History of Animals and other works where he showed naturalist leanings, more empirical works that focused on biological causation and the diversity of life. Aristotle's successor at the Lyceum, wrote a series of books on botany that survived as the most important contribution of antiquity to the plant sciences into the Middle Ages. Scholars of the medieval Islamic world who wrote on biology included al-Jahiz, Al-Dīnawarī, who wrote on botany, Rhazes who wrote on anatomy and physiology.
Medicine was well studied by Islamic scholars working in Greek philosopher traditions, while natural history drew on Aristotelian thought in upholding a fixed hierarchy of life. Biology began to develop and grow with Anton van Leeuwenhoek's dramatic improvement of the microscope, it was that scholars discovered spermatozoa, bacteria and the diversity of microscopic life. Investigations by Jan Swammerdam led to new interest in entomology and helped to develop the basic techniques of microscopic dissection and staining. Advances in microscopy had a profound impact on biological thinking. In the early 19th century, a number of biologists pointed to the central importance of the cell. In 1838, Schleiden and Schwann began promoting the now universal ideas that the basic unit of organisms is the cell and that individual cells have all the characteristics of life, although they opposed the idea that all cells come from the division of other cells. Thanks to the work of Robert Remak and Rudolf Virchow, however, by the 1860s most biologists accepted all three tenets of what came to be known as cell theory.
Meanwhile and classification became the focus of natural historians. Carl Linnaeus published a basic taxonomy for the natural world in 1735, in the 1750s introduced scientific names for all his species. Georges-Louis Leclerc, Comte de Buffon, treated species as artificial categories and living forms as malleable—even suggesting the possibility of common descent. Although he was opposed to evolution, Buffon is a key figure in the history of evolutionary thought. Serious evolutionary thinking originated with the works of Jean-Baptiste Lamarck, the first to present a coherent theory of evolution, he posited that evolution was the result of environmental stress on properties of animals, meaning that the more and rigorously an organ was used, the more complex and efficient it would become, thus adapting the animal to its environment. Lamarck believed that these acquired traits could be passed on to the animal's offspring, who would
Corkscrew (Cedar Point)
Corkscrew is a steel roller coaster built by Arrow Development at Cedar Point in Sandusky, United States. When built in 1976, it was the first roller coaster in the world with 3 inversions; the coaster, which features Arrow's first vertical loop, was built during the same time period as The Great American Revolution at Magic Mountain. However, Revolution opened seven days prior and is therefore credited as the first modern-day coaster to feature a vertical loop; the ride's station is located on the midway directly across from Top Thrill Dragster and was the first coaster to have inversions featuring a walkway underneath. The ride consists of an elevated station, which houses three 24 passenger trains painted red, white & blue, a color scheme inspired by the U. S. Bicentennial in 1976, the year the ride was introduced. Riders leave the station; the slight decline of the station allows the car to roll out down a small drop and around a declining 180 degree curve until the train reaches the chain lift.
The lift operates at or near a speed of 5 mph and ascends, at a 30-degree angle, an 85-foot lift hill. The next element is a bunny hop, so named for fall; the end of this hill is lower than the beginning. It goes through a vertical loop, it coasts up an incline. After a piece of flat track, the train curves around a 180-degree descending turn, heading into the twin corkscrews over the midway of the park. In the "eye" of the corkscrews, it is traveling at 38 mph, it completes these two corkscrew loops, comes through trim and block brakes, coasts back into the station. The ride is 2,050 feet long, consisting of blue tubular steel track with a 48-inch separation between tubes, built on 5 acres, rides for 1 minute and 40 seconds, has three 24-passenger trains. Daily, a train is transferred off the track once ridership reaches a point that permits two-train operation with little or no waiting in line. A different train is cycled off each day; the ride was built by Arrow Dynamics. The total cost of construction was US$1,750,000, the ride has had over 30 million total riders since opening in May 1976.
Cedarpoint.com - Official Corkscrew Page POV of Corkscrew Corkscrew at The Point Online
A stairway, stairwell, flight of stairs, or stairs, is a construction designed to bridge a large vertical distance by dividing it into smaller vertical distances, called steps. Stairs round, or may consist of two or more straight pieces connected at angles. Special types of stairs include ladders; some alternatives to stairs are elevators and inclined moving walkways as well as stationary inclined sidewalks. A stair, or a stairstep, is one step in a flight of stairs. In buildings, stairs is a term applied to a complete flight of steps between two floors. A stair flight is a run of steps between landings. A staircase or stairway is one or more flights of stairs leading from one floor to another, includes landings, newel posts, handrails and additional parts. A stairwell is a compartment extending vertically through a building. A stair hall is the stairs, hallways, or other portions of the public hall through which it is necessary to pass when going from the entrance floor to the other floors of a building.
Box stairs are stairs built between walls with no support except the wall strings. Stairs may be in a straight run, leading from one floor to another without a turn or change in direction. Stairs may change direction by two straight flights connected at a 90 degree angle landing. Stairs may return onto themselves with 180 degree angle landings at each end of straight flights forming a vertical stairway used in multistory and highrise buildings. Many variations of geometrical stairs may be formed of circular and irregular constructions. Stairs may be a required component of egress from buildings. Stairs are provided for convenience to access floors, roofs and walking surfaces not accessible by other means. Stairs may be a fanciful physical construct such as the stairs that go nowhere located at the Winchester Mystery House. Stairs are a subject used in art to represent real or imaginary places built around impossible objects using geometric distortion, as in the work of artist M. C. Escher. "Stairway" is a common metaphor for achievement or loss of a position in the society.
Each step is composed of riser. Tread The part of the stairway, stepped on, it is constructed to the same specifications as any other flooring. The tread "depth" is measured from the outer edge of the step to the vertical "riser" between steps; the "width" is measured from one side to the other. Riser The vertical portion between each tread on the stair; this may be missing for an "open" stair effect. Nosing An edge part of the tread that protrudes over the riser beneath. If it is present, this means that, measured horizontally, the total "run" length of the stairs is not the sum of the tread lengths, as the treads overlap each other. Many building codes require stair nosings for commercial, industrial, or municipal stairs as they provide anti-slip properties and increase pedestrians safety. Starting step or Bullnose Where stairs are open on one or both sides, the first step above the lower floor or landing may be wider than the other steps and rounded; the balusters form a semicircle around the circumference of the rounded portion and the handrail has a horizontal spiral called a "volute" that supports the top of the balusters.
Besides the cosmetic appeal, starting steps allow the balusters to form a wider, more stable base for the end of the handrail. Handrails that end at a post at the foot of the stairs can be less sturdy with a thick post. A double bullnose can be used. Stringer, Stringer board or sometimes just String The structural member that supports the treads and risers in standard staircases. There are three stringers, one on either side and one in the center, with more added as necessary for wider spans. Side stringers are sometimes dadoed to receive treads for increased support. Stringers on open-sided stairs are called "cut stringers". Winders Winders are steps, they are used to change the direction of the stairs without landings. A series of winders form a spiral stairway; when three steps are used to turn a 90° corner, the middle step is called a kite winder as a kite-shaped quadrilateral. Trim Various moldings are used in some instances support stairway elements. Scotia or quarter-round are placed beneath the nosing to support its overhang.
A decorative step at the bottom of the staircase which houses the volute and volute newel turning for a continuous handrail. The balustrade is the system of railings and balusters that prevents people from falling over the edge. Banister, Railing or Handrail The angled member for handholding, as distinguished from the vertical balusters which hold it up for stairs that are open on one side; the term "banister" is sometimes used to mean just the handrail, or sometimes the handrail and the balusters or sometimes just the balusters. Volute A handrail end element for the bullnose step that curves inward like a spiral. A volute is said to be right or left-handed depending on which side of the stairs the handrail is as one faces up the stairs. Turnout Instead of a complete spiral volute, a turnout is a quarter-turn rounded end to the handrail. Gooseneck The vertical handrail that joins a sloped handrail to a higher handrail on the balcony or landing is a
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space; when n = 3, the set of all such locations is called three-dimensional Euclidean space. It is represented by the symbol ℝ3; this serves as a three-parameter model of the physical universe. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions, any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space. Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height and length. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates.
Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are labeled x, y, z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane. Four distinct points can either coplanar or determine the entire space. Two distinct lines can either be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes are parallel. Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane. A hyperplane is a subspace of one dimension less than the dimension of the full space; the hyperplanes of a three-dimensional space are the two-dimensional subspaces. In terms of cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, so, are coplanar. A sphere in 3-space consists of the set of all points in 3-space at a fixed distance r from a central point P.
The solid enclosed by the sphere is called a ball. The volume of the ball is given by V = 4 3 π r 3. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution; the plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane, perpendicular to the axis, is a circle. Simple examples occur. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular cylinder.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0, where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloidThe degenerate quadric surfaces are the empty set, a single point, a single li
In mathematics, a differentiable manifold is a type of manifold, locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts known as an atlas. One may apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas.
The maps that relate the coordinates defined by the various charts to one another are called transition maps. Differentiability means different things in different contexts including: continuously differentiable, k times differentiable and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, differentiable tensor and vector fields. Differentiable manifolds are important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, Yang–Mills theory, it is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus; the study of calculus on differentiable manifolds is known as differential geometry.
The emergence of differential geometry as a distinct discipline is credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen, he motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, presciently described the role of coordinate systems and charts in subsequent formal developments: Having constructed the notion of a manifoldness of n dimensions, found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude... – B. RiemannThe works of physicists such as James Clerk Maxwell, mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one, invariant with respect to coordinate transformations; these ideas found a key application in Einstein's theory of general relativity and its underlying equivalence principle.
A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. A presentation of a topological manifold is a second countable Hausdorff space, locally homeomorphic to a vector space, by a collection of homeomorphisms called charts; the composition of one chart with the inverse of another chart is a function called a transition map, defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced contains the data of how it has been patched together. However, different atlases may produce "the same" manifold. Thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below. There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions.
Some common examples include the following: A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. More a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable. A smooth manifold or C∞-manifold is a differentiable manifold for which all the transition maps are smooth; that is, derivatives of all orders exist. An equivalence class of such atlases is said to be a smooth structure. An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is convergent and equals the function on some open ball. A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic. While there is a meaningful notion of a Ck atlas, there is no distinct notion of a Ck manifold other than C0 and C∞, because for every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure – a result of Whitney.
In fact, eve