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I've been reading the book Gauge, Fields, Knots and Gravity by Baez.

A tangent vector at $p \in M$ is defined as function $V$ from $C^{\infty}(M) $ to $\mathbb R$ satisfying the following properties:

  1. $V(f+g)=V(f) + V(g)$.

  2. $V(\alpha f)= \alpha V(F)$.

  3. $V(fg) = V(f)g(p) + V(g)f(p)$.

Can someone explain me what is the physical interpretation of tangent vectors and the above definition?

  • 3
    This one often causes bewilderment. The advantage of defining tangent vectors this way is that it is very easy to state without introducing local coordinates. As Zhen Lin points out, this definition corresponds to the geometric one by thinking of directional derivatives, i.e. the action of a tangent vector on a function is the directional derivative of the function in that direction.2011-10-24

3 Answers 3