In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots...
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for orthogonal coordinates. The nine gij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g11=h1h1...
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Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There is also a third system, based on two poles...
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Coordinate system (redirect from Coordinates)
intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar...
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Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version...
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Cartesian coordinate system (redirect from Cartesian orthogonal coordinate system)
axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms...
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Unit vector (section Orthogonal coordinates)
{z}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}} They form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra...
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Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude...
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{\mathbf {v} }}_{j}.} Del Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates Griffiths, David J. (2012)...
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coordinates V . {\displaystyle V.} Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates...
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Material derivative (section Orthogonal coordinates)
by the upper convected time derivative. It may be shown that, in orthogonal coordinates, the j-th component of the convection term of the material derivative...
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Orthonormal basis (redirect from Orthogonal set)
basis can be used to define normalized orthogonal coordinates on V . {\displaystyle V.} Under these coordinates, the inner product becomes a dot product...
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Spherical coordinate system (redirect from Spherical coordinates)
perpendicular (orthogonal) to the zenith direction, and typically is desiginated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a...
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of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates. Skew coordinates...
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an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. The...
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equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving...
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means limits application to Cartesian coordinates, in fact most of the common coordinates systems are orthogonal, including familiar ones like cylindrical...
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Differential geometry of surfaces (redirect from Geodesic polar coordinates)
geodesic polar coordinates discussed below, the images of lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If H = (EG)1⁄2...
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Parallel Coordinates plots are a common method of visualizing high-dimensional datasets to analyze multivariate data having multiple variables, or attributes...
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Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional...
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Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about...
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Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate...
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Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals...
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Elliptic coordinate system (redirect from Elliptic coordinates)
the coordinates ( μ , ν ) {\displaystyle (\mu ,\nu )} by substituting the scale factors into the general formulae found in orthogonal coordinates. An...
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Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {\displaystyle (\mu ,\nu ,\lambda )} that generalize two-dimensional...
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Gradient (section Cartesian coordinates)
other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). We consider general coordinates, which...
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{\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}} For all orthogonal coordinates the metric is given by: [ g i j ] = ( h 1 2 0 0 0 h 2 2 0 0 0 h...
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Frame of reference (redirect from Spacetime coordinates)
curvilinear coordinates. Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system. If the basis vectors are orthogonal at...
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Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system...
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Centripetal force (section Polar coordinates)
follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous...
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