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I wonder what is the meaning of the second derivative or what kind of object it is when we have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$.

The first derivative is the Jacobian matrix, but then, what is the second derivative? How can I treat them when I write $f''$ or $D^2 f$?

Thanks a lot for your help!

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    The second derivative is something very big and complicated! :-)2012-12-12
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    More seriously, the second derivative of a mapping like that is an object with 3 indices, a kind of "cubic matrix". Tensor algebra could help in giving it a more precise name, but still, it is a complicated object.2012-12-12
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    Roughly speaking, the second derivative is a tensor.2012-12-12

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