In mathematics, a differential operator is an operator defined as a function of the differentiation operator. (Def: http://en.m.wikipedia.org/wiki/Differential_operator)

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Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
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Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and $$\frac{d}{dx}f(...
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Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
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How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
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What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
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Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\...
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How to prove $(0,1)$ form is not $\overline\partial$-exact

On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
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Differential operator and kernel

Let $P$ a polynomial of two variables, say over the field of real numbers. We define $\partial P$ as $P(\partial_x,\partial_y)$. In this question, it has been shown that if $P_0(x,y)=x^2+y^2$ and $\...
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When do Harmonic polynomials constitute the kernel of a differential operator?

Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...
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Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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What is the meaning of $1/(D+a)$, where $D$ is the derivative operator?

Today I read the answer to this post, in which the poster integrates $x^5e^x$ by making these manipulations with the differential operator $D$: $$\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(1-D+D^2+...)x^5$$...
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differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
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Linear transformations in infinite dimensional vector spaces

If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis ...
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What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} \frac{f(x+...
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Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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How to properly apply the Lie Series

I am trying to solve this problem from Symmetry Methods for Differential Equations A Beginner's Guide (Peter E. Hydon). Use the Lie Series $$F(\hat{x},\hat{y})=\sum_{j=0}^{\infty}\frac{\varepsilon^j}{...
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Is it possible to decompose this expression?

Is it possible to factorize $$(-\partial^2+\phi^2(r))^2-\left(\frac{\partial\phi(r)}{\partial r}\right)^2,$$ where $\phi(r)$ is a function of $r\equiv\sqrt{x^2+y^2+z^2+\xi^2}$? Note that we can not ...
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Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
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Pseudo-differential operators

What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\; \sigma (Q)=q,\;\;\; \sigma (P)=p$ if the series on right side is infinite? ...
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Find the extended form of the group generated by an operator?

I tried to find the extended form of the group generated by the following operators. (I): The first operator $$A=z\frac{\partial }{\partial z}+1$$ To find the extended form of the group ...
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Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
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Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At least,...
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Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
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Complex differential geometric form of the Grothendieck–Hirzebruch–Riemann–Roch theorem

From the wikipedia article, it seems that there should be a differential geometric form of the Grothendieck-Riemann-Roch theorem with schemes replaced by complex manifolds and quasi-coherent sheaves ...
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Proving continuity on spaces of distributions?

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones. When you have a linear operator $T:\mathcal{D}'(\Omega)\...
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Find $a_{n,i,j}$ in the expansion $(x + D)^n = \sum\limits_{i,j} a_{n,i,j} x^i D^j.$

This is problem 47c. in Stanley's Enumerative Combinatorics Vol. 1. Background: Let $D$ be the operator $\frac{d}{dx}$. Part (a) asks to prove $$ (xD)^n = \sum\limits_{k = 0}^n S(n,k)x^k D^k $$ ...
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How to prove convergent function imply its derivative equals to zero?

Let $f\colon (0,\infty) \to\Bbb R$ be differentiable and let $A$ and $B$ be real numbers. Prove that if $f(t) \to A$ and $f′(t) \to B$ as $t \to \infty$ then $B = 0$.
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Simple proof of Chain Rule through $\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$

In an online lecture (link to Youtube), the professor proves the Chain Rule using the following statement: $$\frac{\Delta y}{\Delta x} = \frac{dy}{dx}\biggr|_{x=x_1} + k$$ $$\Delta y = \frac{dy}{dx}\...
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Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
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Taylor expansion of $\frac{1}{1-D}$, where $D$ is the differential operator

I understand that we can represent $e^D$ simply as a power series of D. But what about functions of D which are not entire on the complex plane? What if the function has no taylor expansion, or if it ...
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Divergence of stress tensor in momentum transfer equation

Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of $\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u}) I \...
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Composition of pseudo-differential operators

Denote by $S^m$ the set of functions $p: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that $p \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and $$\left| \frac{\partial^{\alpha+\beta}}{\...
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What does it mean for the leading symbol of a differential operator to be scalar?

I would like to better understand what it means for the leading symbol of a differential operator to be scalar. Concretely, I am currently looking at the Laplace - Beltrami operator on an n-...
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Given $g$ find an $f$ which is solution for $L f = g$. How do I do this?

I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ ...
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Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
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Metric tensors and linear (differential) operators defined on a manifold and its osculating sphere

Given a 1D Riemannian manifold $\Gamma$ embedded in 2D Euclidean space (e.g. a parametric curve on a plane $\mathbb{R}^{2}$ ), and point $x_{0}\in \Gamma$, we denote $S^{1}(x_{0})$ the circle ...
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Is it possible to construct a 1-D linear differential operator with given spectrum $0\leq\lambda_0\leq \lambda_1\leq\dots\leq\lambda_n\le\dots$?

Suppose one is given with a sequence $S$ of non-negative real numbers $0\leq\lambda_0\leq \lambda_1\leq\dots\leq \lambda_n\leq\dots$. Under what conditions on $S$, is it possible to construct a Linear ...
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Applications of Microfunctions

Can anyone suggest good (a) uses/applications or (b) construction of micro-functions (introduced by Mikio Sato in 1971) in analysis? I am trying to understand the subject better. Suggestions of ...
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Space of Differential Operators

Is is possible to talk about a "space of differential operators"? If one defined such a space would it be possible to talk about limits? I don't really have much background so I'm really just looking ...
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Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \frac{...
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Differential operators with arbitrary functions?

By Taylor expansion, one has $$f(x+t) = \sum_{k=0}^∞ \frac{D^k}{k!}f(x)([x+t]-x)^k = \sum_{k=0}^∞ \frac{(Dt)^k}{k!}f(x)$$ and hence one could say $e^{Dt}$ is translation by $t$. But this isn't a "...
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Question about differential operators

Say $N = ab$. How can I express $\frac{d}{dN}$ in terms of $\frac{d}{da}$ and $\frac{d}{db}$?
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About Second-order Linear Homogenous ODE

One way to solve second-order linear homogeneous ode with constant coefficients is to do the following things: $$a\left(\frac{\mathrm d^2}{\mathrm dx^2}\right)f+b\left(\frac{\mathrm d}{\mathrm dx}\...
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How to calculate the derivative of logarithm of a matrix?

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$ The derivative of $\...
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Can anyone prove this equation? (Eq. with operators)

I am trying to understand the last equation from page 2 of this pdf http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap7.pdf but I am not being able to develop as here it says. Could ...
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What is the significance of the integral of the Hessian determinant?

The integral of a function over some region measures the total value of the function in that region: $$T(u)=\int u\thinspace\mathrm{d}V$$ The integral of the squared norm of the gradient of the ...
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Subset of differential operators is a finitely generated module?

I was reading about differential operators, and there is a small claim I don't understand. First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra $...
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Understanding Operators in context of Green's function derivation

I am trying to understand what operators actually mean when deriving the definition of green's function. What is the interagl representation of an operator? Is this correct? $$ D = <x|\int D|x> ...