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An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
The general Stokes theorem applies to higher degree differential forms instead of just 0-forms such as . A closed interval [,] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points and .
When the outer Soddy circle has negative curvature, its center is the isoperimetric point of the triangle: the three triangles formed by this center and two vertices of the starting triangle all have the same perimeter. [4] Triangles whose outer Soddy circle degenerates to a straight line with curvature zero have been called "Soddyian triangles ...
A free group on at least two generators is not boundedly generated (see below). The group SL 2 (Z) is not boundedly generated, since it contains a free subgroup with two generators of index 12. A Gromov-hyperbolic group is boundedly generated if and only if it is virtually cyclic (or elementary), i.e. contains a cyclic subgroup of finite index.
This is considered one of the simplest unsteady problems that has an exact solution for the Navier–Stokes equations. [1] [2] In turbulent flow, this is still named a Stokes boundary layer, but now one has to rely on experiments, numerical simulations or approximate methods in order to obtain useful information on the flow.
In geometry, the incenter–excenter lemma is the theorem that the line segment between the incenter and any excenter of a triangle, or between two excenters, is the diameter of a circle (an incenter–excenter or excenter–excenter circle) also passing through two triangle vertices with its center on the circumcircle.
The two isogonic centers are the intersections of three vesicae piscis whose paired vertices are the vertices of the triangle. When the largest angle of the triangle is not larger than 120°, X(13) is the Fermat point. The angles subtended by the sides of the triangle at X(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1).
An area formula for spherical triangles analogous to the formula for planar triangles. Given a fixed base , an arc of a great circle on a sphere, and two apex points and on the same side of great circle , Lexell's theorem holds that the surface area of the spherical triangle is equal to that of if and only if lies on the small-circle arc , where and are the points antipodal to and , respectively.