Properties of Differential Operators
Differentiation is linear, i.e.,
where f and g are functions, and a is a constant.
Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule
Some care is then required: firstly any function coefficients in the operator D2 must be differentiable as many times as the application of D1 requires. To get a ring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be commutative: an operator gD isn't the same in general as Dg. In fact we have for example the relation basic in quantum mechanics:
The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.
The differential operators also obey the shift theorem.
Read more about this topic: Differential Operator
Famous quotes containing the words properties of, properties and/or differential:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)