Search results
Results from the WOW.Com Content Network
Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicated double integral, which is typically evaluated using a computer algebra system or approximated numerically. Fortunately, many common surfaces form exceptions, and their areas are explicitly known.
Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in ...
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 ...
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
A sphere is the surface of a solid ball, here having radius r. In mathematics, a surface is a mathematical model of the common concept of a surface.It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line.
Implicit non-algebraic surface (wineglass). In mathematics, an implicit surface is a surface in Euclidean space defined by an equation (,,) = An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z.
Given a closed curve in E 3, find a surface having the curve as boundary with minimal area. Such a surface is called a minimal surface. In 1776 Jean Baptiste Meusnier showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface: