In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

In classical geometry, the tangent line to the graph of the function f at a real number a was the unique line through the point ( a, f ( a )) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph.

After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics.

Some classical construction problems of geometry are impossible using compass and straightedge, but can be solved using origami.

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first-and second-order equations, e. g., y

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light.

The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed.

Phenomena that in classical mechanics are ascribed to the action of the force of gravity ( such as free-fall, orbital motion, and spacecraft trajectories ), correspond to inertial motion within a curved geometry of spacetime in general relativity ; there is no gravitational force deflecting objects from their natural, straight paths.

The term is also used as a collective term for the approach to classical, computational and relativistic geometry that makes heavy use of such algebras.

In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

In general, separability is a technical hypothesis on a space which is quite useful and — among the classes of spaces studied in geometry and classical analysis — generally considered to be quite mild.

Many theorems from classical geometry hold true for this spherical geometry as well, but many do not ( see parallel postulate ).

In classical geometry, axioms are general statements while postulates are statements about geometrical objects.

In classical geometry, a proposition may be a construction that satisfies given requirements ; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

Using this, it becomes relatively easy to answer such classical problems of geometry as

The design studies for precast concrete units or for the moulds for in situ shuttering, prompted by the need to obtain a large number of different forms from the combination of a very limited number of units contributed, in the 1980s, to the Taller ’ s affirmation of the validity of classical forms and geometry in contemporary architecture.

Experience is not seen, as it is in classical rationalism, as presenting us initially with clear and distinct objects simply located in space and registering their character, movements, and changes on the tabula rasa of an uninvolved intellect.

One had to manage the given subjects, three diverse recent events, so as to make them part of a classical frieze, -- that is, a pattern of large figures filling the space, with not much else, against a blank background.

-- On the basis of a differentiability assumption in function space, it is possible to prove that, for materials having the property that the stress is given by a functional of the history of the deformation gradients, the classical theory of infinitesimal viscoelasticity is valid when the deformation has been infinitesimal for all times in the past.

These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a Scholium at the very beginning of Newton's work, The Mathematical Principles of Natural Philosophy ( 1687 ), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

Decoherent interpretations of many-worlds using einselection to explain how a small number of classical pointer states can emerge from the enormous Hilbert space of superpositions have been proposed by Wojciech H. Zurek.

There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology.

The classical definition of a sheaf begins with a topological space X.

Today the notion of " absolute space " is abandoned, and an inertial frame in the field of classical mechanics is defined as:

It turns out that they mostly fall into four infinite families, the " classical Lie algebras " A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub > and D < sub > n </ sub >, which have simple descriptions in terms of symmetries of Euclidean space.

So the only similarity of this relativistic aether concept with the classical aether models lies in the presence of physical properties in space.

The most sophisticated example of this is the Sommerfeld – Wilson – Ishiwara quantization rule, which was formulated entirely on the classical phase space.

They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed.

A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations.

A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof.

It starts with the classical algebra of all ( smooth ) functionals over the configuration space.

The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity ( like a classical particle with no forces acting on it ).

A classical field is a function defined over some region of space and time.

The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accomodates the observations of quantum mechanics.

This classical Kelvin – Stokes theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂ Σ:

The so-called classical groups are subgroups of GL ( V ) which preserve some sort of bilinear form on a vector space V. These include the

and can be found via a classical vector coupling model or a more detailed quantum mechanical calculation to be:

In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector ( vector in four-dimensional spacetime ) that replaces classical velocity ( a three-dimensional vector ).

The classical velocity of an object is a tangent vector to its path.

and u is the Euclidean norm of the classical velocity vector:

Quantum information specifies the complete quantum state vector ( or equivalently, wavefunction ) of a system, whereas classical information, roughly speaking, only picks out a definite ( pure ) quantum state if we are already given a prespecified set of distinguishable ( orthogonal ) quantum states to choose from ; such a set forms a basis for the vector space of all the possible pure quantum states ( see pure state ).

In this framework, because one of the observed properties of the electric field was that it was irrotational, and one of the observed properties of the magnetic field was that it was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential ( Coulomb's electrostatic potential, entirely analogous, mathematically, to the classical gravitational potential ) and a stationary magnetic field as the curl of a vector potential ( then a new concept-the idea of a scalar potential was already well accepted by analogy with gravitational potential ).

In classical mechanics, the state of an ideal gas of energy U, volume V and with N particles, each particle having mass m, is represented by specifying the momentum vector p and the position vector x for each particle.

:: Here, as in the classical theory V is a braided vector space of dimension n spanned by the E ´ s, and σ ( a so-called cocylce twist ) creates the nontrivial linking between E ´ s and F ´ s.

A further classical example is the space of lines in projective space of three dimensions ( equivalently, the space of two-dimensional subspaces of a four-dimensional vector space ).

Ricci and Levi-Civita ( following ideas of Elwin Bruno Christoffel ) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold.

In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold.

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system.

The Numerical Recipes books cover a range of topics that include both classical numerical analysis ( interpolation, integration, linear algebra, differential equations, and so on ), signal processing ( Fourier methods, filtering ), statistical treatment of data, and a few topics in machine learning ( hidden Markov models, support vector machines ).

In classical electromagnetism, polarization density ( or electric polarization, or simply polarization ) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material.

In classical mechanics, the Laplace – Runge – Lenz vector ( or simply the LRL vector ) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a planet revolving around a star.

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.