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Darboux Basis (other)
A Darboux basis may refer to: * A Darboux basis of a symplectic vector space * In differential geometry, a Darboux frame on a surface * A Darboux tangent in the dovetail joint {{disambig Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Vector Space
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Darboux Frame
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux. Darboux frame of an embedded curve Let ''S'' be an oriented surface in three-dimensional Euclidean space E3. The construction of Darboux frames on ''S'' first considers frames moving along a curve in ''S'', and then specializes when the curves move in the direction of the principal curvatures. Definition At each point of an oriented surface, one may attach a unit normal vector in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. If is a curve in , parametrized by arc length, then the Darboux frame of is defined by : \mathbf(s) = \gamma'(s), (the ''unit tangent'') : \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dovetail Joint
A dovetail joint or simply dovetail is a joinery technique most commonly used in woodworking joinery (carpentry), including furniture, cabinets, log buildings, and traditional timber framing. Noted for its resistance to being pulled apart (tensile strength), the dovetail joint is commonly used to join the sides of a drawer to the front. A series of 'pins' cut to extend from the end of one board interlock with a series of 'tails' cut into the end of another board. The pins and tails have a trapezoidal shape. Once glued, a wooden dovetail joint requires no mechanical fasteners. History The dovetail joint technique probably pre-dates written history. Some of the earliest known examples of the dovetail joint are in ancient Egyptian furniture entombed with mummies dating from First Dynasty, the tombs of Chinese emperors, and a stone pillar at the Vazhappally Maha Siva Temple in India. The dovetail design is an important method of distinguishing various periods of furniture. The et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
