The Pantheon of Derivatives - Part III - Comments

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There is also a simple example of a nontrivial tangent bundle. Namely it is ##TS^2##; here ##S^2## is the two dimensional sphere. If ##TS^2=\mathbb{R}^2\times S^2## then there exists a vector field ##v(x)## such that ##v(x)\ne 0## for all ##x\in S^2##. But we know that this is not true

And another pretty thing is: why does physics need all these objects ? Simplest example: In classical mechanics the Lagrangian ##L=L(q,\dot q)## is defined on a tangent bundle of configuration manifold while the Hamiltonian ##H=H(q,p)## is a function of cotangent bundle of the configuration manifold: ##L:TM\to\mathbb{R},\quad H:T^*M\to \mathbb{R}##
 
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