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...

intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar...

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...

Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There is also a third system, based on two poles...

coordinates V . {\displaystyle V.} Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates...

Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version...

{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...

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...

{\mathbf {v} }}_{j}.} Del Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates Griffiths, David J. (2012)...

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...

basis can be used to define normalized orthogonal coordinates on V . {\displaystyle V.} Under these coordinates, the inner product becomes a dot product...

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional...

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...

orthogonal, as in orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the metric...

Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals...

means limits application to Cartesian coordinates, in fact most of the common coordinates systems are orthogonal, including familiar ones like cylindrical...

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate...

perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a...

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude...

the coordinates ( μ , ν ) {\displaystyle (\mu ,\nu )} by substituting the scale factors into the general formulae found in orthogonal coordinates. An...

In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional...

Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate...

Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {\displaystyle (\mu ,\nu ,\lambda )} that generalize two-dimensional...

in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates: ( curl ⁡ F )...

{\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...

equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving...

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system...

numerical methods. Orthogonal trajectories are used in mathematics, for example as curved coordinate systems (i.e. elliptic coordinates) and appear in physics...

an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. The...

other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). We consider general coordinates, which...