Surface (topology)In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
Direct currentDirect current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.
Electric currentAn electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge carriers, which may be one of several types of particles, depending on the conductor. In electric circuits the charge carriers are often electrons moving through a wire. In semiconductors they can be electrons or holes.
Current clampIn electrical and electronic engineering, a current clamp, also known as current probe, is an electrical device with jaws which open to allow clamping around an electrical conductor. This allows measurement of the current in a conductor without the need to make physical contact with it, or to disconnect it for insertion through the probe. Current clamps are typically used to read the magnitude of alternating current (AC) and, with additional instrumentation, the phase and waveform can also be measured.
Integrable systemIn mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
Frobenius theorem (differential topology)In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields.
Hyperkähler manifoldIn differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Equivalently, a hyperkähler manifold is a Riemannian manifold of dimension whose holonomy group is contained in the compact symplectic group Sp(n).
Contact geometryIn mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.
Brown ratThe brown rat (Rattus norvegicus), also known as the common rat, street rat, sewer rat, wharf rat, Hanover rat, Norway rat and Norwegian rat, is a widespread species of common rat. One of the largest muroids, it is a brown or grey rodent with a head and body length of up to long, and a tail slightly shorter than that. It weighs between . Thought to have originated in northern China and neighbouring areas, this rodent has now spread to all continents except Antarctica, and is the dominant rat in Europe and much of North America.
Dense setIn topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, is dense in if the smallest closed subset of containing is itself.
Laboratory ratLaboratory rats or lab rats are strains of the subspecies Rattus norvegicus domestica which are bred and kept for scientific research. While less commonly used for research than laboratory mice, rats have served as an important animal model for research in psychology and biomedical science. In 18th century Europe, wild brown rats (Rattus norvegicus) ran rampant and this infestation fueled the industry of rat-catching. Rat-catchers would not only make money by trapping the rodents, but also by selling them for food or, more commonly, for rat-baiting.
Rat-baitingRat-baiting is a blood sport that involves releasing captured rats in an enclosed space with spectators betting on how long a dog, usually a terrier, takes to kill the rats. Often, two dogs competed, with the winner receiving a cash prize. It is now illegal in most countries. In 1835, the Parliament of the United Kingdom implemented an act called the Cruelty to Animals Act 1835, which prohibited the baiting of some animals, such as the bull, bear, and other large mammals.
