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Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.
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Given and differentiable
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 two differentiable manifolds
: Given a function f that has values everywhere on the boundary of a region in R < sup > n </ sup >, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary?
Given a differentiable function defined on a bounded open set, the total variation of has the following expression
Given a twice continuously differentiable function f of one real variable, Taylor's theorem for the case n = 1 states that
Given an exact differential equation defined on some simply connected and open subset D of R < sup > 2 </ sup > with potential function F then a differentiable function f with ( x, f ( x )) in D is a solution if and only if there exists real number c so that
Given and manifold
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
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 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 an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define
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 complex hermitian vector bundle V of complex rank n over a smooth manifold M,
Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:
Given a manifold M representing ( continuous / smooth / with certain boundary conditions / etc.
* Given the action of a Lie algebra g on a manifold M, the set of g-invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.
* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.
Given an oriented manifold M of dimension n with fundamental class, and a G-bundle with characteristic classes, one can pair a product of characteristic classes of total degree n with the fundamental class.
Given a manifold with a submanifold, one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to.
A more general class are flat G-bundles with for a manifold F. Given a representation, the flat-bundle with monodromy is given by, where acts on the universal cover by deck transformations and on F by means of the representation.
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.
Given a function on, one may " geometrize " it by taking it to define a new manifold.
Given a statistical manifold, with coordinates given by, one writes for the probability distribution.
Given and one
Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
Given that one cannot be motivated by reason alone, requiring the input of the passions, Hume argued that reason cannot be behind morality.
Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.
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 good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.
Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.
Given our formula φ, we group strings of quantifiers of one kind together in blocks:
Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.
Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.
Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.
Given significant distance from the magnetic poles, one can figure which hand is which using a magnetic compass and the sun.
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 same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.
Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.
Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.
Given the above fact, one can ask:
Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.
# 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 an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.
Given the unusual casting and production demands, Pacific Overtures remains one of the least-performed musicals by Stephen Sondheim.
Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):
0.279 seconds.