These polar unit vectors can be expressed in terms of Cartesian unit vectors in the x and y directions, denoted i and j respectively :< ref > Note: unlike the Cartesian unit vectors i and j, which are constant, in polar coordinates the direction of the unit vectors u < sub > r </ sub > and u < sub > θ </ sub > depend on θ, and so in general have non-zero time derivatives .</ ref >

Note that all Jones vectors and matrices on this page assumes that the phase of the light wave is, which is used by Hecht.

Note that by definition, a linear combination involves only finitely many vectors ( except as described in Generalizations below ).

Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin ( definition of vector subspace ).

Note, vectors may point into or out of the sphere.

* π is nondegenerate, that is the space of vectors π ( x ) ξ is dense as x ranges through A and ξ ranges through H. Note that if A has an identity, nondegeneracy means exactly π is unit-preserving, i. e. π maps the identity of A to the identity operator on H.

Note that the ( partial ) derivatives of curves and surfaces are vectors that have a direction and a magnitude.

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 basis vectors of a real BCC lattice and the reciprocal lattice of an FCC resemble each other in direction but not in magnitude.

Note: usually all basis vectors are denoted by e, for this article e is for the standard basis ( Cartesian ) and b is for the curvilinear basis.

Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

More formally, a code,, is-locally decodable, if there exists a probabilistic decoder,, such that ( Note: represents the Hamming distance between vectors and ):

Note that if e < sub > a </ sub > are the basis vectors for the Lie algebra, then are the structure constants of the Lie algebra.

Note that ( x, y ) represents the dot product of vectors x and y here.

Note that for any fixed w the function f ( x ) = F ( w x ) is computable ; thus the universality property states that all computable functions of one variable can be obtained in this fashion.

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

Note that Q ( x, y ) is simply replaced by.

Note that the bilinearity and alternating properties imply anticommutativity, i. e., for all elements x, y in, while anticommutativity only implies the alternating property if the field's characteristic is not 2.

Note that the terms < tt > x </ tt > and < tt > y </ tt > aren't alpha-equivalent, because they are not bound in a lambda abstraction.

Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.

Note that if the preorder is R < sup >+=</ sup >, S / ~ is the set of R-cycle equivalence classes: x ∈ if and only if x = y or x is in an R-cycle with y.

The tree is rotated on the edge joining p with its parent g, then rotated on the edge joining x with p. Note that zig-zig steps are the only thing that differentiate splay trees from the rotate to root method introduced by Allen and Munro prior to the introduction of splay trees.

Note that when N = R, the map dφ < sub > x </ sub >: T < sub > x </ sub > M → R coincides with the usual notion of the differential of the function φ.

Note that in any font the glyph of " x " is not exactly the same as one of the multiplication sign.

Note: 2. x / 3. x ( for Series 600 / 800 ) were developed in parallel with 5. x / 6. x ( for Series 200 / 300 / 400 ), so, for example, 3. x was really contemporary with 6. x.

Note that the requirement d ( T ( x ), T ( y )) < d ( x, y ) for all unequal x and y is in general not enough to ensure the existence of a fixed point, as is shown by the map T: < nowiki >

Note that chemists often write H < sup >+</ sup >( aq ) and refer to the hydrogen ion when describing acid-base reactions but the free hydrogen nucleus, a proton, does not exist alone in water, it exists as the hydronium ion, H < sub > 3 </ sub > O < sup >+</ sup >.

Note that the acid can be the charged species and the conjugate base can be neutral in which case the generalized reaction scheme could be written as HA < sup >+</ sup > H < sup >+</ sup > + A.

Note that the beryllium scale is inverted, so increases on this scale indicate lower < sup > 10 </ sup > Be levels

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

Note that t < sup > 0 </ sup > = 1, ( 1 − t )< sup > 0 </ sup > = 1, and that the binomial coefficient,, also expressed as or is:

Note that at ocean pH most of the CO < sub > 2 </ sub > produced in this reaction is immediately converted back into.

Note the position of the hydroxyl group ( red or green ) on the anomeric carbon relative to the CH < sub > 2 </ sub > OH group bound to carbon 5: they are either on the opposite sides ( α ), or the same side ( β ).

Note that in other fields, a lower division between high and ultra-high vacuum is common, often 10 < sup >− 7 </ sup > Pa.

Note that when this routine is called by < tt > jsr </ tt >, AC3 holds the return address.

< cite id =" demographics "> Note I </ cite >-There were around 300, 000 people () working as electrical engineers in the US ; in Australia, there were around 17, 000 () and in Canada, there were around 37, 000 (), constituting about 0. 2 % of the labour force in each of the three countries.

Note that the Bernoulli numbers are defined as B < sub > n </ sub > = B < sub > n </ sub >( 0 ), and that these vanish for odd n greater than 1.

Note that by reversing the equation, we can swap < math > < </ math > and, for example:

Note: Any nonlinear electronic block driven by two signals with frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub > would generate intermodulation ( mixing ) products.

Note that factors p < sup > 0 </ sup > =

Like all subatomic particles, hadrons are assigned quantum numbers corresponding to the representations of the Poincaré group: J < sup > PC </ sup >( m ), where J is the spin quantum number, P the intrinsic parity ( or P-parity ), and C, the charge conjugation ( or C-parity ), and the particle's mass, m. Note that the mass of a hadron has very little to do with the mass of its valence quarks ; rather, due to mass – energy equivalence, most of the mass comes from the large amount of energy associated with the strong interaction.

Note that if n < sub > 1 </ sub > and n < sub > 2 </ sub > are the same, the antisymmetric expression gives zero, which cannot be a state vector as it cannot be normalized.