Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

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 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 an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

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 two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.

Given an SVD of M, as described above, the following two relations hold:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

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 a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

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 subspaces U and W of a vector space V, then their intersection U ∩ W :=

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω 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.