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INFORMAL DESCRIPTION In Differential Geometry , one can attach to every point ''p'' of a differentiable Manifold a tangent space, a real Vector Space which intuitively contains the possible "directions" in which one can pass through ''p''. The elements of the tangent space are called '''tangent vectors''' at ''p''. All the tangent spaces have the same Dimension , equal to the dimension of the manifold. For example, if the given manifold is a 2- Sphere , one can picture the tangent space at a point as the plane which touches the sphere at that point and is Perpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold of Euclidean Space one can picture the tangent space in this literal fashion. In Algebraic Geometry , in contrast, there is an intrinsic definition of tangent space at a point P of a Variety ''V'', that gives a vector space of dimension at least that of ''V''. The points P at which the dimension is exactly that of ''V'' are called the '''non-singular''' points; the others are '''singular''' points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of ''V'' are those where the 'test to be a manifold' fails. See Zariski Tangent Space . Once tangent spaces have been introduced, one can define on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the Tangent Bundle of the manifold. FORMAL DEFINITIONS There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below. Definition as directions of curves Suppose ''M'' is a C''k'' manifold (''k'' ≥ 1) and ''p'' is a point in ''M''. Pick a , and the Equivalence Class es are known as the tangent vectors of ''M'' at ''p''. The equivalence class of the curve γ is written as γ'(0). The tangent space of ''M'' at ''p'', denoted by T''p''''M'', is defined as the set of all tangent vectors; it does not depend on the choice of chart φ. To define the vector space operations on T''p''''M'', we use a chart φ : ''U'' → R''n'' and define the and can thus be used to transfer the vector space operations from R''n'' over to T''p''''M'', turning the latter into an ''n''-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not. Definition via derivations Suppose ''M'' is a C∞ manifold. A real-valued function ''g'' : ''M'' → R belongs to C∞(''M'') if ''g'' o φ-1 is infinitely often differentiable for every chart φ : ''U'' → R''n''. C∞(''M'') is a real Associative Algebra for the Pointwise Product and sum of functions and scalar multiplication. Pick a point ''p'' in ''M''. A '' Derivation '' at ''p'' is a Linear Map ''D'' : C∞(''M'') → R which has the property that for all ''g'', ''h'' in C∞(''M''): D modeled on the Product Rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space T''p''''M''. The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is ''D''(''g'') = (''g'' o γ)'(0) (where the derivative is taken in the ordinary sense, since ''g'' o γ is a function from (-1,1) to R''n''). Definition via the cotangent space Again we start with a C∞ manifold ''M'' and a point ''p'' in ''M''. Consider the Ideal ''I'' in C∞(''M'') consisting of all functions ''g'' such that ''g''(''p'') = 0. Then ''I'' and ''I'' 2 are real vector spaces, and T''p''''M'' may be defined as the Dual Space of the Quotient Space ''I'' / ''I'' 2. This latter quotient space is also known as the Cotangent Space of ''M'' at ''p''. While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the Varieties considered in Algebraic Geometry . If ''D'' is a derivation, then ''D''(''g'') = 0 for every ''g'' in ''I''2, and this means that ''D'' gives rise to a linear map ''I'' / ''I''2 → R. Conversely, if ''r'' : ''I'' / ''I''2 → R is a linear map, then ''D''(''g'') = ''r''((''g'' - ''g''(''p'')) + ''I'' 2) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space. PROPERTIES If ''M'' is an open subset of R''n'', then ''M'' is a C∞ manifold in a natural manner (take the charts to be the Identity Map s), and the tangent spaces are all naturally identified with R''n''. Tangent vectors as directional derivatives One way to think about tangent vectors is as ''directional derivatives''. Given a vector ''v'' in R''n'' one defines the directional derivative of a smooth map ''f'' : R''n''→R at a point ''p'' by |
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