Limiting case (mathematics) in the context of "Electricity and magnetism"

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity ${\displaystyle e}$, a number ranging from ${\displaystyle e=0}$ (the limiting case of a circle) to ${\displaystyle e=1}$ (the limiting case of infinite elongation, no longer an ellipse but a parabola).

An ellipse has a simple algebraic solution for its area, but for its perimeter (also known as circumference), integration is required to obtain an exact solution.

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. If B is true, one can immediately deduce that A is true as well, and if B is false, A can also be immediately deduced to be false. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class; "degeneracy" is the condition of being a degenerate case.

The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a degenerate triangle if at least one side length or angle is zero. Equivalently, it becomes a line segment.

In economics, the Leontief production function or fixed proportions production function is a production function that implies the factors of production which will be used in fixed (technologically predetermined) proportions, as there is no substitutability between factors. It was named after Wassily Leontief and represents a limiting case of the constant elasticity of substitution production function.

For the simple case of a good that is produced with two inputs, the function is of the form

In physics and engineering, heat flux or thermal flux, sometimes also referred to as heat flux density, heat-flow density or heat-flow rate intensity, is a flow of energy per unit area per unit time. Its SI units are watts per square metre (W/m). It has both a direction and a magnitude, and so it is a vector quantity. To define the heat flux at a certain point in space, one takes the limiting case where the size of the surface becomes infinitesimally small.

Heat flux is often denoted ${\displaystyle {\vec {\phi }}_{\mathrm {q} }}$, the subscript q specifying heat flux, as opposed to mass or momentum flux. Fourier's law is an important application of these concepts.

In mathematics, the Gaussian or ordinary hypergeometric function ₂F₁(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.