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There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent space is and why are they equivalent.Most importantly I am having problems connecting the viwepoints.Please help me. Here is link to tangent space definition

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Please include some more information about your background, e.g., is there any particular approach you do understand, or is the problem more connecting these different points of view with each other? –  KCd Jan 13 '13 at 6:10
    
my problem is the second one ie. connecting the viewpoints –  Koushik Jan 13 '13 at 6:12
    
@KCd,what's your full name.I had a teacher whose shortened form of name was KCd –  Koushik Jan 13 '13 at 6:17
    
I am not your former teacher; outside stack exchange I don't use anything like KCd as a name. –  KCd Jan 13 '13 at 6:21

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The "intuition" behind tangent spaces is purely multi-variable calculus. The only new aspect introduced by differential geometry is the desire to apply the methods of multi-variable calculus in more general contexts: e.g. to take our knowledge of calculus on the Euclidean plane and apply it to calculus done on the unit sphere or a torus or some other two-dimensional manifold.

All of those things are familiar constructions from multi-variable calculus.

  1. If $f$ is a curve, then $f'(0)$ is a column-vector.
  2. If $v$ is a column-vector, then $\nabla_v$ is the directional derivative in its direction
  3. If $w$ is a row-vector, then $wv$ is a scalar.
  4. Directional derivatives have the properties of derivations

The tangent space to any point of $\mathbb{R}^n$ is just $\mathbb{R}^n$ again, viewed as the set of column vectors. (the tangent bundle is $\mathbb{R}^n \times \mathbb{R}^n$, the space that would contain the graph of any vector field)

All four of the technical definitions you cite are simply trying to pick out what a vector should be based on its properties. e.g. if you know the value $\omega v$ for every covector $\omega$, that's enough to figure out what $v$ is; and every point in the dual space to the cotangent space corresponds to a vector.

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∇v correcponds to the "direction" directional derivative is taken which again corresponds to the "direction" f′(0)?? –  Koushik Jan 13 '13 at 6:39
    
i don't understand what you say about covector –  Koushik Jan 13 '13 at 6:40

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