Given any coordinate chart about some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:

Given a local coordinate system x < sup > i </ sup > on the manifold, the reference axes for the coordinate system are the vector fields

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

Given a DMS ( Degrees, Minutes, Seconds ) coordinate such as W87 ° 43 ′ 41 ″, convert it to a number of decimal degrees using the following method:

Given a MinDec ( Degrees, Minutes, Decimal Minutes ) coordinate such as 40 ° 26. 7717N, 79 ° 56. 93172W, convert it to a number of decimal degrees using the following method ( for example for 79 ° 56. 93172W ):

Given a decimal longitudinal coordinate such as-87. 728055 it will be necessary to know whether it is a latitudinal or longitudinal coordinate in order to fully convert it.

Given coordinate functions

Using tensor calculus, proper time is more rigorously defined in general relativity as follows: Given a spacetime which is a pseudo-Riemannian manifold mapped with a coordinate system and equipped with a corresponding metric tensor, the proper time experienced in moving between two events along a timelike path P is given by the line integral

: Given such a tangent vector v, let f be the curve given in the x < sup > i </ sup > coordinate system by.

Given these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

Given the system ( corresponding to )

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given that the bulk of family chores fell to women, and that they were five times more likely than men to head a single-parent family, the shortcomings of Finland's child day-care system affected women more than it did men.

Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given a syscall that does nothing, a full round-trip under BSD would require about 40μs, whereas on a user-space Mach system it would take just under 500μs.

Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.

Given the increasing number of scientists who study the nervous system, several prominent neuroscience organizations have been formed to provide a forum to all neuroscientists and educators.

Given the complexity of NLP problems, it is often difficult to predict performance only on the basis of glass-box evaluation, but this type of evaluation is more informative with respect to error analysis or future developments of a system.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given the opportunity, for example during an illness that suppresses the immune system, the virus is reactivated and travels to the end of the nerve cell, where it causes the symptoms described above.

: Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates.

Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

) Given that the telephone system nominal voiceband ( including guardband ) is 4, 000 Hz, the required digital sampling rate is 8, 000 Hz ( see Nyquist rate ).

Given the structural similarities between bacterial flagella and bacterial secretory systems, it is thought that bacterial flagella may have evolved from the type three secretion system ; however, it is not known for certain whether these pores are derived from the bacterial flagella or the bacterial secretory system.

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given a Riemannian manifold with metric tensor, we can compute the Ricci tensor, which collects averages of sectional curvatures into a kind of " trace " of the Riemann curvature tensor.

Given a specified distribution of matter and energy in the form of a stress – energy tensor, the EFE are understood to be equations for the metric tensor, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.

Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.

is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ( M, I ), it is always equipped with a compatible hyperkähler metric.

Given the frame field, one can also define a metric by conceiving of the frame field as an orthonormal vector field.

Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution.

Given the metric η, we can ignore the covariant and contravariant distinction for T.

Given a metric space ( X, d ), or more generally, an extended pseudoquasimetric ( which will be abbreviated xpq-metric here ), one can define an induced map d: X × P ( X )→ by d ( x, A ) = inf

Given a metric space a point is called close or near to a set if

Given a metric on a Lorentzian manifold, the Weyl tensor for this metric may be computed.

Given two tangent vectors u and v at a point x in M, the metric can be evaluated on u and v to give a real number:

Given a manifold M, one looks for the longest product of systoles which give a " curvature-free " lower bound for the total volume of M ( with a constant independent of the metric ).

Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner: