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A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
The set of the zeros of a function of three variables is a surface, which is called an implicit surface. [1] If the defining three-variate function is a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise. [31] [32] Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified ...
The area of a ring is π(R 2 − r 2), where R is the outer radius (in this case f(y)), and r is the inner radius (in this case g(y)). The volume of each infinitesimal disc is therefore πf(y) 2 dy. The limit of the Riemann sum of the volumes of the discs between a and b becomes integral (1).
where g is the determinant of the first fundamental form of the surface mapping r(s, t). [1] [2] For example, if we want to find the surface area of the graph of some scalar function, say z = f(x, y), we have
A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort. For instance: The sphere and a disc both have genus zero. A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee ...
In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space into its field of scalars (that is, it is an element of the dual space) [1] In functional analysis and related fields, it refers to a mapping from a space X {\displaystyle X} into the field of real or complex numbers .
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