1.
Regular polytope
–
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, Regular polytopes are the generalized analog in any number of dimensions of regular polygons and regular polyhedra. The strong symmetry of the regular polytopes gives them an aesthetic quality that interests both non-mathematicians and mathematicians, classically, a regular polytope in n dimensions may be defined as having regular facets and regular vertex figures. These two conditions are sufficient to ensure that all faces are alike and all vertices are alike, note, however, that this definition does not work for abstract polytopes. A regular polytope can be represented by a Schläfli symbol of the form, with regular facets as, Regular polytopes are classified primarily according to their dimensionality. They can be classified according to symmetry. For example, the cube and the regular octahedron share the same symmetry, indeed, symmetry groups are sometimes named after regular polytopes, for example the tetrahedral and icosahedral symmetries. Three special classes of regular polytope exist in every dimensionality, Regular simplex Measure polytope Cross polytope In two dimensions there are many regular polygons. In three and four dimensions there are more regular polyhedra and 4-polytopes besides these three. In five dimensions and above, these are the only ones, see also the list of regular polytopes. The idea of a polytope is sometimes generalised to include related kinds of geometrical object, some of these have regular examples, as discussed in the section on historical discovery below. A concise symbolic representation for regular polytopes was developed by Ludwig Schläfli in the 19th Century, the notation is best explained by adding one dimension at a time. A convex regular polygon having n sides is denoted by, so an equilateral triangle is, a square, and so on indefinitely. A regular star polygon which winds m times around its centre is denoted by the fractional value, a regular polyhedron having faces with p faces joining around a vertex is denoted by. The nine regular polyhedra are and. is the figure of the polyhedron. A regular 4-polytope having cells with q cells joining around an edge is denoted by, the vertex figure of the 4-polytope is a. A five-dimensional regular polytope is an, the dual of a regular polytope is also a regular polytope. The Schläfli symbol for the dual polytope is just the original written backwards, is self-dual, is dual to, to
2.
Force
–
In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
3.
Spacetime
–
In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum. Until the turn of the 20th century, the assumption had been that the 3D geometry of the universe was distinct from time, Einsteins theory was framed in terms of kinematics, and showed how measurements of space and time varied for observers in different reference frames. His theory was an advance over Lorentzs 1904 theory of electromagnetic phenomena. A key feature of this interpretation is the definition of an interval that combines distance. Although measurements of distance and time between events differ among observers, the interval is independent of the inertial frame of reference in which they are recorded. The resultant spacetime came to be known as Minkowski space, non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. Mathematically, spacetime is a manifold, which is to say, by analogy, at small enough scales, a globe appears flat. An extremely large scale factor, c relates distances measured in space with distances measured in time, waves implied the existence of a medium which waved, but attempts to measure the properties of the hypothetical luminiferous aether implied by these experiments provided contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the speed of light in air plus the speed of the flowing water, the partial aether-dragging implied by this result was in conflict with measurements of stellar aberration. By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein were to derive later, but with a fundamentally different interpretation. As a theory of dynamics, his theory assumed actual physical deformations of the constituents of matter. For example, most physicists believed that Lorentz contraction would be detectable by such experiments as the Trouton–Noble experiment or the Experiments of Rayleigh and Brace. However, these negative results, and in his 1904 theory of the electron. Einstein performed his analyses in terms of kinematics rather than dynamics and it would appear that he did not at first think geometrically about spacetime. It was Einsteins former mathematics professor, Hermann Minkowski, who was to provide an interpretation of special relativity. Einstein was initially dismissive of the interpretation of special relativity
4.
Electromagnetism
–
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as fields, magnetic fields. The other three fundamental interactions are the interaction, the weak interaction, and gravitation. The word electromagnetism is a form of two Greek terms, ἤλεκτρον, ēlektron, amber, and μαγνῆτις λίθος magnētis lithos, which means magnesian stone. The electromagnetic force plays a role in determining the internal properties of most objects encountered in daily life. Ordinary matter takes its form as a result of forces between individual atoms and molecules in matter, and is a manifestation of the electromagnetic force. Electrons are bound by the force to atomic nuclei, and their orbital shapes. The electromagnetic force governs the processes involved in chemistry, which arise from interactions between the electrons of neighboring atoms, there are numerous mathematical descriptions of the electromagnetic field. In classical electrodynamics, electric fields are described as electric potential, although electromagnetism is considered one of the four fundamental forces, at high energy the weak force and electromagnetic force are unified as a single electroweak force. In the history of the universe, during the epoch the unified force broke into the two separate forces as the universe cooled. Originally, electricity and magnetism were considered to be two separate forces, Magnetic poles attract or repel one another in a manner similar to positive and negative charges and always exist as pairs, every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding magnetic field outside the wire. Its direction depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a field, or a magnet is moved towards or away from it. While preparing for a lecture on 21 April 1820, Hans Christian Ørsted made a surprising observation. As he was setting up his materials, he noticed a compass needle deflected away from north when the electric current from the battery he was using was switched on. At the time of discovery, Ørsted did not suggest any explanation of the phenomenon. However, three later he began more intensive investigations
5.
Screw theory
–
Screw theory is the algebra and calculus of pairs of vectors, such as forces and moments and angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms. The pair of vectors that form the Plücker coordinates of a line define a unit screw, an important result of screw theory is that geometric calculations for points using vectors have parallel geometric calculations for lines obtained by replacing vectors with screws. This is termed the transfer principle, screw theory has become an important tool in robot mechanics, mechanical design, computational geometry and multibody dynamics. This is in part because of the relationship between screws and dual quaternions which have used to interpolate rigid-body motions. Based on screw theory, an efficient approach has also developed for the type synthesis of parallel mechanisms. Fundamental theorems include Poinsots theorem and Chasles theorem, felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program. He also worked out elliptic geometry, and a view of Euclidean geometry. The use of a matrix for a von Staudt conic and metric. Other prominent contributors include Julius Plücker, W. K. Clifford, F. M. Dimentberg, Kenneth H. Hunt, J. R. Phillips. A spatial displacement of a body can be defined by a rotation about a line. This is known as Chasles theorem, for comparison, the six parameters that define a spatial displacement can also be given by three Euler Angles that define the rotation and the three components of the translation vector. The components of the screw define the Plücker coordinates of a line in space, the force and torque vectors that arise in applying Newtons laws to a rigid body can be assembled into a screw called a wrench. A force has a point of application and a line of action, a torque, on the other hand, is a pure moment that is not bound to a line in space and is an infinite pitch screw. The ratio of these two magnitudes defines the pitch of the screw, a twist represents the velocity of a rigid body as an angular velocity around an axis and a linear velocity along this axis. All points in the body have the same component of the velocity along the axis, Thus, the helicoidal field formed by the velocity vectors in a moving rigid body flattens out the further the points are radially from the twist axis. The points in a body undergoing a constant screw motion trace helices in the fixed frame, if this screw motion has zero pitch then the trajectories trace circles, and the movement is a pure rotation. If the screw motion has infinite pitch then the trajectories are all lines in the same direction
6.
Rotations in 4-dimensional Euclidean space
–
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO. The name comes from the fact that it is the orthogonal group of order 4. In this article rotation means rotational displacement, for the sake of uniqueness rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise. A fixed plane is a plane for which every vector in the plane is unchanged after the rotation, an invariant plane is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation. Four-dimensional rotations are of two types, simple rotations and double rotations, a simple rotation R about a rotation centre O leaves an entire plane A through O fixed. Every plane B that is orthogonal to A intersects A in a certain point P. Each such point P is the centre of the 2D rotation induced by R in B, all these 2D rotations have the same rotation angle α. Half-lines from O in the axis-plane A are not displaced, half-lines from O orthogonal to A are displaced through α, all other half-lines are displaced through an angle < α. For each rotation R of 4-space, there is at least one pair of orthogonal 2-planes A and B each of which are invariant, hence R operating on either of these planes produces an ordinary rotation of that plane. For almost all R, the rotation angles α in plane A and β in plane B — both assumed to be nonzero — are different, the unequal rotation angles α and β satisfying -π < α, β < π are almost* uniquely determined by R. Assuming that 4-space is oriented, then the orientations of the 2-planes A and B can be consistent with this orientation in two ways. If the rotation angles are unequal, R is sometimes termed a double rotation, *Assuming that 4-space is oriented, then an orientation for each of the 2-planes A and B can be chosen to be consistent with this orientation of 4-space in two equally valid ways. If the angles from one choice of orientations of A and B are. If the rotation angles of a rotation are equal then there are infinitely many invariant planes instead of just two, and all half-lines from O are displaced through the same angle. Such rotations are called isoclinic or equiangular rotations, or Clifford displacements, beware, not all planes through O are invariant under isoclinic rotations, only planes that are spanned by a half-line and the corresponding displaced half-line are invariant. Assuming that an orientation has been chosen for 4-dimensional space. Now assume that only the rotation angle α is specified, then there are in general four isoclinic rotations in planes OUX and OYZ with rotation angle α, depending on the rotation senses in OUX and OYZ. We make the convention that the senses from OU to OX
7.
Rotation
–
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
8.
Manifold
–
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, 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 exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and 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. 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 exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
9.
Magnetic field
–
A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
10.
Phase diagram
–
For the use of this term in mathematics and physics, see phase space. Common components of a phase diagram are lines of equilibrium or phase boundaries, phase transitions occur along lines of equilibrium. Triple points are points on phase diagrams where lines of equilibrium intersect, triple points mark conditions at which three different phases can coexist. For example, the phase diagram has a triple point corresponding to the single temperature and pressure at which solid, liquid. The solidus is the temperature below which the substance is stable in the solid state, the liquidus is the temperature above which the substance is stable in a liquid state. There may be a gap between the solidus and liquidus, within the gap, the substance consists of a mixture of crystals, the simplest phase diagrams are pressure–temperature diagrams of a single simple substance, such as water. The axes correspond to the pressure and temperature, the phase diagram shows, in pressure–temperature space, the lines of equilibrium or phase boundaries between the three phases of solid, liquid, and gas. The fusion curves for water and Antimony are with negative slopes, the curves on the phase diagram show the points where the free energy becomes non-analytic, their derivatives with respect to the coordinates change discontinuously. For example, the capacity of a container filled with ice will change abruptly as the container is heated past the melting point. The open spaces, where the energy is analytic, correspond to single phase regions. Single phase regions are separated by lines of non-analytical behavior, where phase transitions occur, in the diagram on the left, the phase boundary between liquid and gas does not continue indefinitely. Instead, it terminates at a point on the diagram called the critical point. This reflects the fact that, at high temperatures and pressures. In water, the critical point occurs at around Tc =647.096 K, pc =22.064 MPa, the existence of the liquid–gas critical point reveals a slight ambiguity in labelling the single phase regions. Thus, the liquid and gaseous phases can blend continuously into each other, the solid–liquid phase boundary can only end in a critical point if the solid and liquid phases have the same symmetry group. Thus, the substance requires a temperature for its molecules to have enough energy to break out of the fixed pattern of the solid phase. A similar concept applies to liquid–gas phase changes, water, because of its particular properties, is one of the several exceptions to the rule. In addition to temperature and pressure, other properties may be graphed in phase diagrams
11.
Non-Euclidean geometry
–
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. In elliptic geometry the lines curve toward each other and intersect, the debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. Other mathematicians have devised simpler forms of this property, regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
12.
Vector field
–
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as, the elements of differential and integral calculus extend naturally to vector fields. Vector fields can usefully be thought of as representing the velocity of a flow in space. In coordinates, a field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point. More generally, vector fields are defined on manifolds, which are spaces that look like Euclidean space on small scales. In this setting, a field gives a tangent vector at each point of the manifold. Vector fields are one kind of tensor field, given a subset S in Rn, a vector field is represented by a vector-valued function V, S → Rn in standard Cartesian coordinates. If each component of V is continuous, then V is a vector field. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space, in physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a distinct entity from a simple list of scalars. Thus, suppose that is a choice of Cartesian coordinates, in terms of which the components of the vector V are V x =, then the components of the vector V in the new coordinates are required to satisfy the transformation law Such a transformation law is called contravariant. Given a differentiable manifold M, a field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that p ∘ F is the identity mapping where p denotes the projection from TM to M, in other words, a vector field is a section of the tangent bundle. If the manifold M is smooth or analytic—that is, the change of coordinates is smooth —then one can make sense of the notion of vector fields. The collection of all vector fields on a smooth manifold M is often denoted by Γ or C∞. A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind, the length of the arrow will be an indication of the wind speed