Vector calculusVector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations.
Final topologyIn general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous. The quotient topology on a quotient space is a final topology, with respect to a single surjective function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps.
Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
TangentIn geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f(c), where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
Tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is, where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at . There is a natural projection defined by . This projection maps each element of the tangent space to the single point .
Fusion rocketA fusion rocket is a theoretical design for a rocket driven by fusion propulsion that could provide efficient and sustained acceleration in space without the need to carry a large fuel supply. The design requires fusion power technology beyond current capabilities, and much larger and more complex rockets. Fusion nuclear pulse propulsion is one approach to using nuclear fusion energy to provide propulsion. Fusion's main advantage is its very high specific impulse, while its main disadvantage is the (likely) large mass of the reactor.
Ruled surfaceIn geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle.
Magnetic mirrorA magnetic mirror, known as a magnetic trap (магнитный захват) in Russia and briefly as a pyrotron in the US, is a type of magnetic confinement fusion device used in fusion power to trap high temperature plasma using magnetic fields. The mirror was one of the earliest major approaches to fusion power, along with the stellarator and z-pinch machines. In a classic magnetic mirror, a configuration of electromagnets is used to create an area with an increasing density of magnetic field lines at either end of a confinement volume.
Euler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by (Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.
Solenoidal vector fieldIn vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: A common way of expressing this property is to say that the field has no sources or sinks. The divergence theorem gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero: where is the outward normal to each surface element.
PolywellThe polywell is a design for a fusion reactor based on two ideas: heating ions by concentrating (-) charge to accelerate the ions and trapping a diamagnetic plasma inside a cusp field. This kind of plasma trap is based on the idea that a plasma will create its own magnetic field that rejects the outside field. That is not common behavior for a fusing plasma. A similar trapping concept was tested by the Lockheed-Martin high beta fusion reactor team, which tried to hold a plasma in a similar way.
Complex lamellar vector fieldIn vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl. A lamellar vector field is a special case given by vector fields with zero curl. The adjective "lamellar" derives from the noun "lamella", which means a thin layer.
