The Tangent Space

  • Klaus Jänich
Part of the Undergraduate Texts in Mathematics book series (UTM)


One of the basic ideas of differential calculus is to approximate differentiable maps by linear maps so as to reduce analytic (hard) problems to linear-algebraic (easy) problems whenever possible. Recall that locally at x, the linear approximation of a map f: ℝ n → ℝ k is the differential df x , ℝ n → ℝ k of f at x. The differential is characterized by \(f\left( {x + v} \right) + f\left( x \right) + d{f_x}\cdot v + \varphi \left( v \right)\), where \(\mathop {\lim }\limits_{v \to 0} \varphi \left( v \right)/\left\| v \right\| = 0\), and given by the Jacobian matrix. But how can a differentiable map f: MN between. manifolds be characterized locally at pM by a linear map?


Jacobian Matrix Tangent Space Tangent Vector Real Vector Space Differential Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Klaus Jänich
    • 1
  1. 1.NWF-I MathematikUniversität RegensburgRegensburgGermany

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