# The Tangent Space

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

## Abstract

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?

## Keywords

Jacobian Matrix Tangent Space Tangent Vector Real Vector Space Differential Calculus
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