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Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its ...
is a smooth section of the projection map; we say that ω is a smooth differential m-form on M along f −1 (y). Then there is a smooth differential (m − n)-form σ on f −1 (y) such that, at each x ∈ f −1 (y), = /. This form is denoted ω / η y.
The Cartesian (x′,y′) axes are related to the rotated graticule in the same way that the axes (x,y) axes are related to the standard graticule. The tangent transverse Mercator projection defines the coordinates ( x′ , y′ ) in terms of − λ′ and φ′ by the transformation formulae of the tangent Normal Mercator projection:
The two U(1) factors can be combined into U(1) Y × U(1) l, where l is the lepton number. Gauging of the lepton number is ruled out by experiment, leaving only the possible gauge group SU(2) L × U(1) Y. A similar argument in the quark sector also gives the same result for the electroweak theory.
The Poincaré lemma states that if B is an open ball in R n, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n. [1] More generally, the lemma states that on a contractible open subset of a manifold (e.g., ), a closed p-form, p > 0, is exact. [citation needed]
In each zone the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce a secant projection with two standard lines, or lines of true scale, about 180 km on each side of, and about parallel to, the central meridian (Arc cos 0.9996 = 1.62° at the Equator). The scale is less than 1 inside the standard ...
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Each triple (,,) determines a line, the line determined is unchanged if it is multiplied by a non-zero scalar, and at least one of , and must be non-zero. So the triple ( s , t , u ) {\displaystyle (s,t,u)} may be taken to be homogeneous coordinates of a line in the projective plane, that is line coordinates as opposed to point coordinates.