orthogonal coordinates वाक्य
उदाहरण वाक्य
मोबाइल
- CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.
- However, there are other orthogonal coordinate systems in three dimensions that cannot be obtained by projecting or rotating a two-dimensional system, such as the ellipsoidal coordinates.
- Since basis vectors generally vary in orthogonal coordinates, if two vectors are added whose components are calculated at different points in space, the different basis vectors require consideration.
- Another approach, which goes in line with ideas of differential geometry and conformal geometry, is orthogonal coordinates, where coordinate hypersurfaces of different coordinates are orthogonal, although curved.
- So the projection onto the eigenvectors of L is simply an orthogonal coordinate transformation of the initial condition to a set of coordinates which decay exponentially and independently of each other.
- The domain for these equations is commonly a 3 or less Euclidean space, for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar partial derivative equations to be solved.
- But this must simplify in some way also since the flow is assumed 2D . If orthogonal coordinates are assumed, the curl takes on a fairly simple form, and the equation above expanded becomes:
- While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.
- A simple method for generating orthogonal coordinates systems in two dimensions is by a conformal mapping of a standard two-dimensional grid of Cartesian coordinates ( " x ", " y " ).
- But for the higher order terms ( the two coming from the divergence of the deviatoric stress that distinguish Navier Stokes equations from Euler equations ) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems.
- For example, most of the three-dimensional sets of orthogonal coordinates are derived from either projecting or rotating a two-dimensional orthogonal coordinate system; hence, the rotated ones all include a form of polar coordinates as a subset.
- For example, most of the three-dimensional sets of orthogonal coordinates are derived from either projecting or rotating a two-dimensional orthogonal coordinate system; hence, the rotated ones all include a form of polar coordinates as a subset.
- If orthogonal coordinates are used, there would be scale factors along the diagonal part of the spacelike part of the metric, while for general curvilinear coordinates the entire spacelike part of the metric would have components dependent on the curvilinear basis used.
- The six independent scalar products " g ij " = "'h "'i . "'h "'j of the natural basis vectors generalize the three scale factors defined above for orthogonal coordinates.
- This is " not " true for general bases : orthogonal coordinates have diagonal metrics containing various scale factors ( i . e . not necessarily 1 ), while general curvilinear coordinates could also have nonzero entries for off-diagonal components.
- Is the Jacobian determinant, which has the geometric interpretation of the deformation in volume from the infinitesimal cube d " x " d " y " d " z " to the infinitesimal curved volume in the orthogonal coordinates.
- Terse notation for the cross product, which simplifies generalization to non-orthogonal coordinates and higher dimensions, is possible with the Levi-Civita tensor, which will have components other than zeros and ones if the scale factors are not all equal to one.
- Orthogonal coordinates in three and higher dimensions can be generated from an orthogonal two-dimensional coordinate system, either by projecting it into a new dimension ( " cylindrical coordinates " ) or by rotating the two-dimensional system about one of its symmetry axes.
- Other differential operators such as \ nabla \ cdot \ mathbf { F } and \ nabla \ times \ mathbf { F } can be expressed in the coordinates ( \ sigma, \ tau ) by substituting the scale factors into the general formulae found in orthogonal coordinates.
- Other differential operators such as \ nabla \ cdot \ mathbf { F } and \ nabla \ times \ mathbf { F } can be expressed in the coordinates ( ?, ?, ? ) by substituting the scale factors into the general formulae found in orthogonal coordinates.
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