Differential formIn mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f: Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S: The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms.
ParticleIn the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion.
Particle physicsParticle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three generations of fermions, although ordinary matter is made only from the first fermion generation. The first generation consists of up and down quarks which form protons and neutrons, and electrons and electron neutrinos.
Differential geometryDifferential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.
Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
11 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.
Subatomic particleIn physics, a subatomic particle is a particle smaller than an atom. According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a proton, neutron, or meson), or an elementary particle, which is not composed of other particles (for example, an electron, photon, or muon). Particle physics and nuclear physics study these particles and how they interact.
Elementary particleIn particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles, twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number electrons and other leptons, quarks, and the fundamental bosons.
Closed and exact differential formsIn mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the of d, and a closed form is in the kernel of d. For an exact form α, α = dβ for some differential form β of degree one less than that of α. The form β is called a "potential form" or "primitive" for α.
Primary educationPrimary education or elementary education is typically the first stage of formal education, coming after preschool/kindergarten and before secondary school. Primary education takes place in primary schools, elementary schools, or first schools and middle schools, depending on the location. The International Standard Classification of Education considers primary education as a single-phase where programmes are typically designed to provide fundamental reading, writing, and mathematics skills and establish a solid foundation for learning.
Primary schoolA primary school (in Ireland, India, the United Kingdom, Australia, New Zealand, Trinidad and Tobago, Jamaica, and South Africa), elementary school, or grade school (in North America and the Philippines) is a school for primary education of children who are 4 to 10 years of age (and in many cases, 11 years of age). Primary schooling follows pre-school and precedes secondary schooling.
Complex differential formIn mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures.
