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From Wikipedia:

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative.

I think the $n$-th order (total) derivative of a mapping is the derivative of its $n-1$-th order derivative. In other words, derivative can be defined recursively. So I was wondering how to understand that "the natural analog of second, third, and higher-order total derivatives ... is not built by repeatedly taking the total derivative"?

Thanks and regards!

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    I just thought the same thing about that Wikipedia article. As it is it looks plain wrong and Didier's answer below corroborates this.2012-03-21

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