In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Then, we sort the list by coordinates, and update the matrix as we go.

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

Then any linear functional can be written in these coordinates as a sum of the form:

Then in the coordinates metric reads:

Then one loads the molecular structures ' Cartesian coordinates ( e. g. from PDB files ).

Then it was close to the ideal positions near at the celestial equator of the sky coordinates.

Then the dipole moment will be zero, and if we also scale the coordinates so that the points are at unit distance from the center, in opposite direction, the system's quadrupole moment will then simply be

Then in the y coordinates, the 1-jet of v is a new list of real numbers.

Then, φ decomposes into the sum of a field configuration with no punctures, φ < sub > 0 </ sub > and where we have switched to the complex plane coordinates for convenience.

Then for isotropic rectangular coordinates,,,

Then locally, one may choose canonical coordinates ( q < sup > 1 </ sup >, ..., q < sup > n </ sup >, p < sub > 1 </ sub >, ..., p < sub > n </ sub >) on M, in which the symplectic form is expressed as

Then the control channel coordinates the system so talkgroups can share these frequencies seamlessly.

Let's say a particle in this body has coordinate ( x < sub > 1 </ sub >, y < sub > 1 </ sub >, z < sub > 1 </ sub >) along with x-coordinate ( x < sub > 2 </ sub >) and y-coordinate ( y < sub > 2 </ sub >) of second particle. Then using distance formula of distance between two coordinates we have distance d = sqrt ((( x < sub > 1 </ sub >- x < sub > 2 </ sub >)< sup > 2 </ sup >+( y < sub > 1 </ sub >- y < sub > 2 </ sub >)< sup > 2 </ sup >+( z < sub > 1 </ sub >- z < sub > 2 </ sub >)< sup > 2 </ sup >))

Then the magnetic field can be written in cartesian coordinates as

Then the coordinates of the d points of intersection are algebraic functions of the Plücker coordinates of U, and by taking a symmetric function of the algebraic functions, a homogeneous polynomial known as the Chow form ( or Cayley form ) of Z is obtained.

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Then consider a vector tangent to:

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then the joint distribution of is multivariate normal with mean vector and covariance matrix

Then any vector in R < sup > 3 </ sup > is a linear combination of e < sub > 1 </ sub >, e < sub > 2 </ sub > and e < sub > 3 </ sub >.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Then the two equations still allow the normal to rotate around the view vector, thus additional constraints are needed from prior geometric information.

Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.

Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.

Then k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set

Then applying the Gram – Schmidt process to the three vectors ( A < sub > 2 </ sub >− A < sub > 1 </ sub >, A < sub > 3 </ sub >− A < sub > 1 </ sub >, V ) produces an orthonormal basis of space, the third vector of which will be normal to plane A.

Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

Then and are equivalent: The functor which maps the object of to the vector space and the matrices in to the corresponding linear maps is full, faithful and essentially surjective.

Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P ( W ) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space ( the subspace

Let E → M be a vector bundle of rank k and let F ( E ) be the principal frame bundle of E. Then a ( principal ) connection on F ( E ) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F ( E ) → R < sup > k </ sup >.

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Then, A is a vector potential for v, that is,

Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then the ban was enforced again by Leo V in 815.

Then W is a subspace of V.

Then the columns of V such that the corresponding form an orthonormal basis of the nullspace of A.

In his last memoir, Hart-Davis listed the books he had edited as: The Second Omnibus Book ( Heinemann ) 1930 ; Then and Now ( Cape ) 1935 ; The Essential Neville Cardus ( Cape ) 1949 ; Cricket All His Life by E. V.

Then followed Sectionum conicarum libri V. ( Edinburgh, 1735 ), a second edition of which, with additions, appeared in 1750.

Then, V < sub > 2 </ sub > ( the safe takeoff speed ) is called.

Using the natural isomorphism between and the space of linear transformations from V to V ,< ref name =" natural iso "> Let L ( V, V ) be the space of linear transformations from V to V. Then the natural map

Let be an-sequence cofinal on and generic over L. Then no set in L of L-size smaller than ( which is uncountable in V, since is preserved ) can cover, since is a regular cardinal.

* Let k be the coordinate ring of the variety V. Then the dimension of V is the transcendence degree of the field of fractions of k over k.