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On Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator.

The link that underlies the words "differentiation operator" in fact gives many different concepts related to differentiation, but no exact definition of the "differentiation operator".

I understand the differential operator acts on functions (over some domain), is linear and satisfies the product rule (is that right?)

So what then is the differentiation operator ?

It cannot be the same, because the Differential Operator is supposed to be a function of it .. just as polynomial in $x$ is not the same as $x$.

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Can you show us the link on Wikipedia you are talking about? The differentiation operator is just the linear operator from the function space of differential functions to the space of continuous functions –  tomcuchta Jan 22 '12 at 21:18
@tomcuchta: I think you mean the function space of differentiable functions. –  joriki Jan 22 '12 at 21:37
@tomcuchta: I have added the link above, the differentiation operator is mentioned right in the introductory paragraph of that link. –  harlekin Jan 22 '12 at 21:39
@joriki Yep! Typo... –  tomcuchta Jan 23 '12 at 2:57
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2 Answers

up vote 6 down vote accepted

There's no such thing as the differential operator, but there is a class of things that are, collectively, called differential operators.

Imagine that we're working in a function space such as $\mathcal C^\infty(\mathbb R)$. Then $\mathcal C^\infty$ is a vector space (by pointwise addition and multiplication by constants), and a linear map from $\mathcal C^\infty$ to itself is called an operator. The space of operators is itself a vector space, and we can make it into a non-commutative ring by declaring "multiplication" of operators to be functional composition (since the composition of two linear maps is itself a linear map).

We can decide on a subring of "simple" operators, such as all operators that multiply by a constant, or all operators that work by pointwise multiplication by a fixed function. Which operators are "simple" depends on the application. (Beware that this sense of "simple" is not standard; I'm defining it for the occasion here).

Now, if our function space is sufficiently well behaved, the differentiation operator $\partial$ (which takes a function to its derivative) is a linear map from the function space to itself, and therefore an operator. A differential operator is then an element of the least subring that contains the simple operators and the differentiation operator. Such an operator is always the value of some polynomial expression with $\partial$ as the variable and simple operators as coefficients, and it is in this sense that your reference considers a differential operator to be a "function of" $\partial$.

In particular, the differentiation operator is itself a differential operator.

Often one would not consider the simple operators themselves to be "differential operator"; but in other cases one might want to consider them trivial cases of differential operators. That, again, depends on the application, and on authorial preference.

In more complex functions spaces, such as $\mathcal C^\infty(\mathbb R^n,\mathbb R)$ one might want to consider several basic differentiation operators, such as $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, and a differential operator is then something that can be written as an expresion involving those. Things get yet more involved on smooth manifolds, but the basic principles stay the same.

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I think part of your misunderstanding is that there is not "the" differential operator – the article describes a differential operator as an operator defined as a function of the differentiation operator. (In case your native language doesn't have articles: a definite article implies that there is exactly one entity suitable as a referent of the subsequent noun phrase, whereas an indefinite article implies that there may be any number of such suitable referents.)

In fact, what the introduction somewhat imprecisely calls "the" differentiation operator can also be one of various things, as evidenced by the first section that shows various shades of differentiation operators, total and partial and with respect to certain variables. However, the idea is that in each case the operator performs a single differentiation operation and nothing else.

By contrast, a differential operator can act by differentiating several times or even as a linear combination such as $\mathrm d^2/\mathrm dx^2+2\mathrm d/\mathrm dx$. I see that Henning has already written an answer that has more on the ring of differential operators, so I'll leave it at that.

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