[permalink] [id link]
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:
from
Wikipedia
Some Related Sentences
Given and two
Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".
Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.
Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.
Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.
Given the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.
Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.
Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.
Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both
Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.
Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).
Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.
: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.
Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.
Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space
Given this understanding, the Board withdrew the allegation of unsportsmanlike behaviour two days before the fourth Test, thus saving the tour.
Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.
Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.
Given the notion of a lexeme, it is possible to distinguish two kinds of morphological rules.
Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.
Given barely two months to assemble a large sea-going invasion fleet, the Kriegsmarine opted to convert inland river barges into makeshift landing craft.
In the account books of Johanna, Duchess of Brabant and Wenceslaus I, Duke of Luxemburg, an entry dated May 14, 1379 reads: " Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards ".
# Given any two distinct points, there is exactly one line incident with both of them.
# Given any two distinct lines, there is exactly one point incident with both of them.
Given and tangent
Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
* Given a point and a circle, to draw either tangent.
Given a circle k, with a center O, and a point P outside of the circle, we want to construct the ( red ) tangent ( s ) to k that pass through P. Suppose the ( as yet unknown ) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent.
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define
Given vector fields and lying in the tangent bundle, the affine connection describes how to differentiate the vector field along the direction.
Given any linear map A on each tangent space of M ; i. e., A is a tensor field of rank ( 1, 1 ), then the Nijenhuis tensor is a tensor field of rank ( 1, 2 ) given by
Given any immersed submanifold S of M, the tangent space to a point p in S can naturally be thought of as a linear subspace of the tangent space to p in M. This follows from the fact that the inclusion map is an immersion and provides an injection
Given three mutually tangent circles (< font color =" black "> black </ font >), what radius can a fourth tangent circle have?
Given a manifold Q, a vector field X on Q ( or equivalently, a section of the tangent bundle TQ ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces.
Given a collection of differential 1-forms α < sub > i </ sub >, i = 1, 2, ..., k on an n-dimensional manifold M, an integral manifold is a submanifold whose tangent space at every point p ∈ M is annihilated by each α < sub > i </ sub >.
: Given such a tangent vector v, let f be the curve given in the x < sup > i </ sup > coordinate system by.
Given and vectors
Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has
Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.
Given the inevitable errors in measuring the intensities, and the mathematical difficulties of reconstructing atomic positions from the interatomic vectors, this technique is rarely used to solve structures, except for the simplest crystals.
Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.
Given a rotating frame composed by three unitary vectors, all the three must have the same angular speed in any instant.
Given the vectors
* Given two complex vectors x and y, multiplication by U preserves their inner product ; that is,
Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix.
Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis.
Given *- representations π, π ' each with unit norm cyclic vectors ξ ∈ H, ξ ' ∈ K such that their respective associated states coincide, then π, π ' are unitarily equivalent representations.
Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.
Given a C < sup > n + 1 </ sup >- curve γ in R < sup > n </ sup > which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors
Given a surface, one may integrate over its scalar fields ( that is, functions which return numbers as values ), and vector fields ( that is, functions which return vectors as values ).
Given a vector space V, the polynomials on this space are S ( V *), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing.
Given an incident direction from the surface to the light source and the surface normal direction the specularly reflected direction ( all unit vectors ) is:
Given an orthonormal basis, any pure state of a two-level quantum system can be written as a superposition of the basis vectors
Given any bilinear form ƒ on V the set of vectors
Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M < sub > 1 </ sub > and M < sub > 2 </ sub >, parametrised by model parameter vectors and is assessed by the Bayes factor K given by
Given vectors a, b and c, the product
Given a possibly non-orthogonal set of vectors and the projection related is
Given a real matrix M and vector q, the linear complementarity problem seeks vectors z and w which satisfy the following constraints:
Case: Given a positive ( or more generally irreducible non-negative matrix ) A, for all non-negative non-zero vectors x and f ( x ) as the minimum value of < sub > i </ sub > / x < sub > i </ sub > taken over all those i such that x < sub > i </ sub > ≠ 0, then f is a real valued function whose maximum is the Perron – Frobenius eigenvalue r.
0.345 seconds.