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From Wikipedia:

Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the derivative, one repeatedly applies partial derivatives with respect to different variables. For example, the second order partial derivatives of a scalar function of n variables can be organized into an n by n matrix, the Hessian matrix. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a tensor). Nevertheless, higher derivatives have important applications to analysis of local extrema of a function at its critical points. For an advanced application of this analysis to topology of manifolds, see Morse theory.

In multivariable calculus, I was told that higher derivatives were tensors and that was the reason we never went beyond Hessians (none of us had studied tensors before). If higher derivatives aren't tensors, then what are they? Where can I learn more about them?

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  • $\begingroup$ Honestly, I've never seen a presentation of tensors that made much sense without considering them as sections of bundles. It's like defining a matrix to be an object that transforms in a certain magical way under a change of basis, rather than considering it as a linear transformation. $\endgroup$
    – anomaly
    May 29, 2015 at 20:36
  • $\begingroup$ @Blex I'm looking for something on higher derivatives of functions of several variables. But info on tensors is appreciated, too. :) $\endgroup$
    – user244460
    May 29, 2015 at 20:40
  • $\begingroup$ @Blex, the link is off $\endgroup$
    – user561334
    Nov 30, 2018 at 3:49
  • $\begingroup$ fixed: If you are looking for something to learn about tensors have a look at this pdf (A Gentle Introduction to tensors), it's a really nice introduction. $\endgroup$
    – Blex
    May 31, 2019 at 11:18

1 Answer 1

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I have no idea what the Wikipedia entry is trying to say, but higher derivatives are perfectly well-defined intrinsic objects. Let $V=\mathbb R^n$ and $W=\mathbb R^m$ and let $f: V \to W$ be a differentiable function. (This also works for Banach spaces in general, with some small modifications.) The derivative of $f$ is the map $$ Df: V \to L(V,W), $$ where $L(V,W)$ denotes the space of linear transformations, which associates to a point in $V$ the Jacobian derivative of $f$ at that point.

Now considering $L(V,W)$ as a (Banach) space in its own right, $f$ is twice-differentiable when $Df$ is differentiable, in which case the second derivative $D^2f$ of $f$ is the derivative of $Df$, which is then a map $$ D^2f: V \to L(V,L(V,W)). $$ Identifying $L(V,L(V,W))$ with the space of bilinear maps from $V \times V$ to $W$ leads thinking of the second derivative as a $2$-tensor. Clearly this process can be continued, leading to higher-order derivatives as well; in particular, the third derivative of $f$ would be a map $$ D^3f: V \to L(V,L(V,L(V,W))), $$ which can be thought of as a $3$-tensor and some identifications.

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  • $\begingroup$ Interestingly, the Wikipedia entry on the Frechet derivative outlines precisely what I say here, so this entry is at odds with the one linked to in the original post. $\endgroup$ May 29, 2015 at 21:22
  • $\begingroup$ It seems to me that Wikipedia's comment comes from manifold theory: higher derivatives don't transform well under change of coordinates. $\endgroup$ May 29, 2015 at 21:26
  • $\begingroup$ Possibly, but then it seems to be misplaced in the section it's in, which is talking about multivariable functions as studied in multivariable calculus. $\endgroup$ May 29, 2015 at 21:28
  • $\begingroup$ I still think it's reasonable to say that the Hessian is not a tensor when it doesn't transform by a pointwise formula under coordinate changes. It's just that in the multivariable calculus case you have a preferred choice of coordinates, so these problems are not as acute. $\endgroup$ May 29, 2015 at 21:30
  • $\begingroup$ @KevinCarlson So if they are not tensors, then what are they? And is there a book I can read to learn more about them? $\endgroup$
    – user244460
    May 29, 2015 at 21:43

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