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.
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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.
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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.
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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.
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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.
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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.
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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:
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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.
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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 " ).
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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.
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