In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces...
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cotangent bundle of the manifold. The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic...
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A similar definition applies to space curves and curves in n-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect...
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A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself...
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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally)...
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tangent space at each event is a vector space of the same dimension as spacetime, 4. In practice, one need not be concerned with the tangent spaces....
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Riemannian manifold (redirect from Riemann space)
metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together...
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Pushforward (differential) (redirect from Tangent map)
is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle \varphi \colon M\to N} is a smooth...
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Normal mapping (section Calculating tangent spaces)
reuse is made possible by encoding maps in tangent space. The tangent space is a vector space, which is tangent to the model's surface. The coordinate system...
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Differentiable manifold (section Definition of tangent space and differentiation in local coordinates)
a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of Rn to this tangent space. A moving...
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of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in...
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In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. In...
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of M fixing p and acting on the tangent space T p M {\displaystyle T_{p}M} as minus the identity (every symmetric space is complete, since any geodesic...
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vector space over that field (the cotangent space). The tangent space T x ( X ) {\displaystyle T_{x}(X)} is defined as the dual of this vector space. The...
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Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian...
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Pseudo-Riemannian manifold (redirect from Pseudo-Riemannian space)
positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity...
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algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction...
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space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space....
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about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable...
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geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions...
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an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X and Dp f is the derivative (pushforward)...
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Orientability (redirect from Orientation (space))
transformation on each tangent space, then the manifold M is orientable. Conversely, M is orientable if and only if the structure group of the tangent bundle can...
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functional on the tangent space at each point. A Riemannian manifold, or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner...
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Tangential and normal components (redirect from Tangent component)
manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to...
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geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold. In...
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vectors from one tangent space to another. Parallel transport of tangent vectors is a way of moving vectors from one tangent space to another along a...
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that the tangent space T Q {\displaystyle TQ} corresponds to the velocities of the points q ∈ Q {\displaystyle q\in Q} , while the cotangent space T ∗ Q...
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The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold...
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to "slide along" that submanifold. They span the tangent space of that submanifold. The tangent space T p N {\displaystyle T_{p}N} should have the same...
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Finsler manifold (redirect from Finsler space)
a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ : [a...
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