Symmetry of Second Derivatives - Formal Expressions of Symmetry

Formal Expressions of Symmetry

In symbols, the symmetry says that, for example,

$\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) = \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right).$

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_i . D_j = D_j . D_i.

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 monomials, so that one can take polynomials in the x_i as a domain. In fact smooth functions are possible.

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