Classical field theoryA classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically intended to describe electromagnetism and gravitation, two of the fundamental forces of nature.
Lagrangian mechanicsFormulation of classical mechanicsJoseph-Louis Lagrange (1736–1813)In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique.
Partial differential equationType of differential equation
Riemannian manifoldSmooth manifold with an inner product on each tangent spaceThe dot product of two vectors tangent to the sphere sitting inside 3-dimensional Euclidean space contains information about the lengths and angle between the vectors.
Field (physics)In science, a field or field quantity is a physical quantity – represented by a scalar, vector, spinor, or tensor – that has a value for each point in space and time. An example of a scalar field is a weather map of surface temperatures, described by assigning a number to each point on the map. A map of surface winds, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field.
Potential theoryIn mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" dates from 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation.
Chern–Gauss–Bonnet theoremIn mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an ).
Quantum field theoryIn theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current Standard Model of particle physics is based on QFT.
Fiber bundleIn mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.Sobolev spaceIn mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n ) {\displaystyle (f_{n})} converges uniformly to a limiting function f {\displaystyle f} if, roughly speaking, they uniformly approximate the function f {\displaystyle f} over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar of the original function.
Degrees of freedom (physics and chemistry)In physics and chemistry, a degree of freedom is an independent physical parameter in the chosen parameterization of a physical system. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smallest number n{\textstyle n} of parameters whose values need to be known in order to always be possible to determine the values of all parameters in the chosen parameterization.*As an Amazon Associate I earn from qualifying purchases.