| Symmetry Of Second Derivatives |
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f of ''n'' variables. If the partial derivative with respect to ''x''''i'' is denoted with a subscript ''i'', then the symmetry is the assertion that f is an ''n'' × ''n'' Symmetric Matrix . This matrix is called the Hessian Matrix of ''f''. The entries in it off the Main Diagonal are the '''mixed derivatives'''; that is, successive partial derivatives with respect to different variables. In most normal circumstances the Hessian matrix is indeed symmetric; but from the point of view of Mathematical Analysis this isn't a safe statement, without some hypothesis on ''f'' that goes further than simply stating the existence of the second derivatives at a particular point. Clairaut's Theorem gives a sufficient condition on ''f'' for this to occur. In symbols, the symmetry says that, for example, : This equality can also be written as : Alternatively, the symmetry can be written as an algebraic statement involving the Differential Operator ''D''''i'' which takes the partial derivative with respect to ''x''''i'': D From this relation it follows that the Ring of differential operators with Constant Coefficients , generated by the ''D''''i'', is Commutative . But one should naturally specify some domain for these operators. It is easy to check the symmetry as applied to Monomial s, so that one can take Polynomial s in the ''x''''i'' as a domain. In fact Smooth Function s is possible. One can also apply the theory of Distribution s to get round any analytic problems with the symmetry. Firstly the derivative of any function can be defined (provided it is Integrable ), as a distribution. Secondly the use of Integration By Parts throws the symmetry question back onto the test functions, which are smooth and certainly satisfy the symmetry. One concludes that, in the sense of distributions, the symmetry always holds. (Another approach, where the Fourier Transform of a function is defined, is to note that on transforms the partial derivatives become multiplication operators that commute much more obviously). The fact remains that in the worst case there are Counterexample s. In the case of two variables, near (0, 0) one can consider two limiting processes on f corresponding to making ''h'' → 0 first, and to making ''k'' → 0 first. These processes need not ''commute'': it can matter, looking at the first-order terms, which is applied first. This leads to the construction of Pathological examples in which the symmetry of second derivatives is not true. someone write up such an example here or in its own article, and add that to the List Of Mathematical Examples ?'' Given that the derivatives as Schwartz distributions are symmetric, this kind of example belongs in the 'fine' theory of Real Analysis . A more sophisticated argument is this: consider the first-order differential operators ''D''''i'' to be Infinitesimal Operator s on Euclidean Space . That is, ''D''''i'' in a sense generates the One-parameter Group of Translation s parallel to the ''x''''i''-axis. These groups certainly all commute with each other, and therefore we expect that the Infinitesimal Generator s do also; the Lie Bracket : ''D''''j'' = 0 is the way that is reflected. When one comes to ask whether this is try as applied to spaces of functions on Euclidean space, this question is an elementary part of the theory of Differentiable Vector s in Representation Theory ; that is, a suitable subspace of a Function Space on which the Lie Algebra representation deriving from a representation of a Lie Group has desirable and transparent properties. COUNTEREXAMPLE Consider the function : with . Then the mixed partial derivatives of ''f'' exist, and are continuous everywhere except at . Moreover : at . |
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