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Area formula (geometric measure theory) In geometric measure theory the area formula relates the Hausdorff measure of the image of a Lipschitz map, while accounting for multiplicity, to the integral of the Jacobian of the map. It is one of the fundamental results of the field that has connections, for example, to rectifiability and Sard's theorem .
Lévy's stochastic area. In probability theory, Lévy's stochastic area is a stochastic process that describes the enclosed area of a trajectory of a two-dimensional Brownian motion and its chord. The process was introduced by Paul Lévy in 1940, [1] and in 1950 [2] he computed the characteristic function and conditional characteristic function.
The Pediatric Risk of Mortality ( PRISM) score was developed from the Physiologic Stability Index (PSI) [1] to reduce the number of physiologic variables required for pediatric intensive-care unit (PICU) mortality risk assessment, from 34 (in the PSI) to 14, [2] and to obtain an objective weighting of the remaining variables. [citation needed ...
Ellipsoid. This is a list of volume formulas of basic shapes: [4] : 405–406. Cone – , where is the base 's radius. Cube – , where is the side's length; Cuboid – , where , , and are the sides' length; Cylinder – , where is the base's radius and is the cone's height; Ellipsoid – , where , , and are the semi-major and semi-minor axes ...
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. [2] It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical ...
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e ...
Heptagonal bipyramid. Properties. Convex semiregular. Vertex figure. 3D model of a (uniform) heptagonal prism. In geometry, the heptagonal prism is a prism with heptagonal base. This polyhedron has 9 faces (2 bases and 7 sides), 21 edges, and 14 vertices. [1] [2]
The first variation of area formula is a fundamental computation for how this quantity is affected by the deformation of the submanifold. The fundamental quantity is to do with the mean curvature. Let (M, g) denote a Riemannian manifold, and consider an oriented smooth manifold S (possibly with boundary) together with a one-parameter family of ...