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Hello,

I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C

Why so?

Don't we simply need a class C

What do we need the partial derivatives of

The Jacobian is made of only first-order partial derivatives after all.

I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C

^{∞}(i.e. a*smooth*manifold).Why so?

Don't we simply need a class C

^{1}differentiable manifold in order to have tangent spaces and do differential geometry?What do we need the partial derivatives of

*all*orders, in particular of third, fourth, fifth order for?The Jacobian is made of only first-order partial derivatives after all.

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