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
Famous quotes containing the words formal, expressions and/or symmetry:
“This is no argument against teaching manners to the young. On the contrary, it is a fine old tradition that ought to be resurrected from its current mothballs and put to work...In fact, children are much more comfortable when they know the guide rules for handling the social amenities. It’s no more fun for a child to be introduced to a strange adult and have no idea what to say or do than it is for a grownup to go to a formal dinner and have no idea what fork to use.”
—Leontine Young (20th century)
“We ought to celebrate this hour by expressions of manly joy. Not thanks, not prayer seem quite the highest or truest name for our communication with the infinite,—but glad and conspiring reception,—reception that becomes giving in its turn, as the receiver is only the All-Giver in part and infancy.”
—Ralph Waldo Emerson (1803–1882)
“What makes a regiment of soldiers a more noble object of view than the same mass of mob? Their arms, their dresses, their banners, and the art and artificial symmetry of their position and movements.”
—George Gordon Noel Byron (1788–1824)