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.
32.
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.
33.
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.
34.
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.
35.
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.
36.
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.
37.
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.
38.
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.
39.
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.
40.
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.
