Vector Analysis pp 25-48 | Cite as

# The Tangent Space

## 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*: *M* → *N* between. *manifolds* be characterized locally at *p* ∈ *M* by a linear map?

## Keywords

Jacobian Matrix Tangent Space Tangent Vector Real Vector Space Differential Calculus## Preview

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