# Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

394 questions
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### High order derivatives on manifold

Suppose I have a Riemannian manifold $M$ and a smooth function $f:M \to \mathbb{R}$. I denote the gradient of $f$ with $\nabla f$. What is the meaning of $$\nabla^N f$$ with $N \ge 3$ integer? Is it ...
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### Exponential differential operator

Consider the operator $D= e^{ax*d/dx}$ operating on an infinitely differentiable function f(x). My approach: $Df(x)= f(x) + ax*df(x)/dx + (ax)^2*d^2f(x)/dx^2 + ...$ $=f(x+ax)$ But this does not ...
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### Convergence to the differential operator

This is not an accurate question. I am not so sure about unbounded operators on a Hilbert space. Let $D$ be the differential operator on $L^2(0,1)$. Well, to somewhat, we can extend $D$ to a normal ...
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### $\exp(a^2\partial_x^2)f(x) = ?$

I can prove that $\exp(a\partial_x)f(x) = f(x+a)$, but what happens for second derivatives? To be more precise, what is the right-hand side of $\exp(a^2\partial_x^2)f(x)$? The above operator has an ...
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### suggest a course on differential operator and Weyl

Can someone suggest me a course in pdf or textbook about differential operator and Weyl algebra. let $R$ a commutative algebra over a field $k$ let M, N two R-modules. we define the set of ...
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### Differential operator commuting with Euclidean transformations

Why is it true that a differential operator $S$ on $\mathbb{R}^n$ commuting with translations and rotations must be of the form $$S = \sum a_j \Delta^j$$ where the $a_j$ are constant coefficients ?
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### Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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### Computing a derivative through Lie series

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where a locally unique solution $x(t)$, starting from the initial condition $x$, is denoted as $x(t)=\phi(t,x)$. Assume ...
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### Proving 2nd ode [closed]

If $x=e^t$, can someone give me proof that $$\frac{d^2}{dx^2}=\frac{1}{e^{2t}}\left(\frac{d^2}{dt^2}−\frac{d}{dt}\right).$$ Thank you