The Pantheon of Derivatives - Part III - Comments

[fresh_42]

fresh_42 submitted a new PF Insights post

The Pantheon of Derivatives - Part III

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[jedishrfu]

These are great articles!

How many more parts?

 

[Greg Bernhardt]

How many more parts?

two

 

[zwierz]

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: $L:TM\to\mathbb{R},\quad H:T^*M\to \mathbb{R}$