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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 ...
In differential geometry, the sign of the area of a region of a surface is associated with the orientation of the surface. [7] The area of a set A in differential geometry is obtained as an integration of a density : μ ( A ) = ∫ A d x ∧ d y , {\displaystyle \mu (A)=\int _{A}dx\wedge dy,} where d x and d y are differential 1-forms that make ...
A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
The graph of a continuous function of two variables, defined over a connected open subset of R 2 is a topological surface. If the function is differentiable, the graph is a differentiable surface. A plane is both an algebraic surface and a differentiable surface. It is also a ruled surface and a surface of revolution.
The simplest type of parametric surfaces is given by the graphs of functions of two variables: = (,), (,) = (,, (,)). A rational surface is a surface that admits parameterizations by a rational function. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its ...
Graph of = /. Gabriel's horn is formed by taking the graph of =, with the domain and rotating it in three dimensions about the x axis. The discovery was made using Cavalieri's principle before the invention of calculus, but today, calculus can be used to calculate the volume and surface area of the horn between x = 1 and x = a, where a > 1. [6]
The second fundamental form of a general parametric surface S is defined as follows. Let r = r(u 1,u 2) be a regular parametrization of a surface in R 3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u α by r α, α = 1, 2.
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry . Analytic geometry is used in physics and engineering , and also in aviation , rocketry , space science , and spaceflight .