Search results
Results from the WOW.Com Content Network
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).
In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.
By a theorem of Descartes, this is equal to 4 π divided by the number of vertices (i.e. the total defect at all vertices is 4 π). The three-dimensional analog of a plane angle is a solid angle . The solid angle, Ω , at the vertex of a Platonic solid is given in terms of the dihedral angle by
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 Stokes I, Q, U and V parameters. The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation.They were defined by George Gabriel Stokes in 1851, [1] [2] as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of ...
This means the bipyramids' vertices correspond to the faces of a prism, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other; doubling it results in the original polyhedron. A triangular bipyramid is the dual polyhedron of a triangular prism, and vice versa.
A triangulated polygon with 11 vertices: 11 sides and 8 diagonals form 9 triangles. Every simple polygon can be partitioned into non-overlapping triangles by a subset of its diagonals. When the polygon has n {\displaystyle n} sides, this produces n − 2 {\displaystyle n-2} triangles, separated by n − 3 {\displaystyle n-3} diagonals.
Sperner's lemma states that, if a big triangle is subdivided into smaller triangles meeting edge-to-edge, and the vertices are labeled with three colors so that only two of the colors are used along each edge of the big triangle, then at least one of the smaller triangles has vertices of all three colors; it has applications in fixed-point ...