Note that these axes will be orthogonal if all entries in the χ tensor are real, corresponding to a case in which the refractive index is real in all directions.

Note that is a tensor with nine terms.

Note that in general, no such relation exists in spaces not endowed with a metric tensor.

: Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases.

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion ; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

( Note: this has three indices, but is not a tensor ).

Note that the tensor field only needs to be defined on the curve for this definition to make sense.

Note that it is not a tensor ( does not transform as a tensor ).

Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but v ⊗ w is not a symmetric 2-tensor.

Note that the transition between " upper " (" contravariant ") and " lower " (" covariant ") vector or tensor components is trivial for a indices ( e. g. ), whereas for μ and ν it is nontrivial, corresponding e. g. to the usual Lorentz signature,.

Note that the Christoffel symbol does not transform as a tensor, but rather as an object in the jet bundle.

Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-reversing coordinate transformations.

Note that is a tensor ( see also tensor product ).

Note too that the trace of the tidal tensor can also be written

Note that the rank of rear admiral is quite different from the honorary office Rear-Admiral of the United Kingdom.

Note that the number of non-zero is exactly the rank of the matrix.

Note that the IDF uses this rank across all three of its services.

Note that although many air forces use the rank of second lieutenant, in most Commonwealth air forces the equivalent rank of pilot officer is used.

* Artesh-bod ( Iran ) ( Note: Since the Islamic Revolution of 1979 no one has been promoted to this rank.

Note: Before unification in 1968 of the Royal Canadian Air Force, Canadian Army, and the Royal Canadian Navy, rank structure and insignia followed the British pattern.

Note the first edition has Fairbairn ’ s rank as ‘ Captain ’ all subsequent ( 1940's ) editions as ‘ Major ’.

Note crimson shoes ( buty karmazynowe ), a sign of wealth and rank ( magnates nickname were ' karmazyni ' - the crimson ones-because of those boots.

Note that the German inscription names his rank as oberst ( colonel ).

Note that the Wehrmacht and SS rank of Generalleutnant was equivalent to the rank of major general in the U. S. and British armies, as well as other western militaries.

Note: The second rank of Lance Corporal, formerly " Private ", was changed in January 2010, due to the French translation " Soldat " being the word " soldier ".

Note: This story is set in the days before Gordon achieved the rank of police commissioner, and Bullock the rank of detective.

Note that there is doubt about rank equivalency in countries that have Fleet Admirals but no Commodores, such as the former German Empire and Russia-often it is considered that in these countries a Fleet Admiral equates to an OF-9 rank, an Admiral to OF-8, a Vice Admiral to OF-7, and a Counter / Rear Admiral to OF-6 ( i. e. the ranks all move down one grade ).

Note that there is doubt about rank equivalency in countries that have no brigadier-generals but both colonel-generals and full generals, such as the German Empire and Russia-often it is considered that in these countries a colonel-general equates to an OF-9 rank, a general to OF-8, a lieutenant-general to OF-7, and a major-general to OF-6 ( i. e. the ranks below colonel-general all move down one grade ).

Note that the implementation as disjoint-set forests doesn't allow deletion of edges — even without path compression or the rank heuristic.

Note: This category treats " General " as an occupation, not a rank.

Note that there is doubt about rank equivalency in countries that have both colonel-generals and generals but no brigadier-generals, such as German Empire and Russia-often it is considered that in these countries a colonel-general equates to an OF-9 rank, a general to OF-8, a lieutenant-general to OF-7, and a major-general to OF-6 ( i. e. the ranks below colonel-general all move down one grade ).

Note that this is before the effects of round-off error are taken into account ; conditioning is a property of the matrix, not the algorithm or floating point accuracy of the computer used to solve the corresponding system.

( Note that there are N independent eigenvectors ; a unitary matrix is never defective.

Note that the trace is only defined for a square matrix ( i. e., n × n ).

Note that the larger grains ( pebbles and gravel ) in the till are completely surrounded by the matrix of finer material ( silt and sand ), and this characteristic, known as matrix support, is diagnostic of till.

Note also that the X < sub > i </ sub > are in general not independent ; they can be seen as the result of applying the matrix A to a collection of independent Gaussian variables z.

Note how the equation above reduces to that of the univariate normal distribution if is a matrix ( i. e. a real number ).

The Julia set of a function ƒ is commonly denoted J ( ƒ ), and the Fatou set is denoted F ( ƒ ).< ref > Note that for other areas of mathematics the notation can also represent the Jacobian matrix of a real valued mapping between smooth manifolds .</ ref > These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.

Note that the matrix defined as above has the following properties:

Note that some books define the Jacobian as the transpose of the matrix given above.

Note that the adjugate is the transpose of the cofactor matrix.

Note that if there are exactly n distinct eigenvalues in an n × n matrix then this matrix is diagonalizable.

Note that the eigenvalues appear in the diagonal matrix.

Note that for any orthogonal matrix if we set and, the criteria for being factors and factor loadings still hold.

Note that if is a Dirac fermion, then the chiral symmetry can be written as where is some matrix acting on.

Note that any matrix obtained by multiplying by a complex scalar λ determines the same transformation, so a Möbius transformation determines its matrix only up to scalar multiples.

Note the two sieves catching charred seeds and charcoal, and the bags of archaeological matrix waiting for flotation.

Note that this does not always mean that the left of the matrix will be an identity matrix.