Formal Expressions of Symmetry
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 Di which takes the partial derivative with respect to xi:
- Di . Dj = Dj . Di.
From this relation it follows that the ring of differential operators with constant coefficients, generated by the Di, 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 xi as a domain. In fact smooth functions are possible.
Read more about this topic: Symmetry Of Second Derivatives
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