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Questions tagged [differential-operators]

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

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Vector Fields as Differential Operators

Given a manifold $\mathcal{M}$, the notion of a vector field $\xi$ on $\mathcal{M}$ can be interpreted as a collection of arrows on the manifold. In the book The Road to Reality by Roger Penrose, ...
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Understanding Partial derivative - how they work in this example

Can someone please help me in understanding why the following equation is true? $[(r\frac{\partial}{\partial r})^2+ r\frac{\partial}{\partial r}] = \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\...
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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
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Definition of the weight $k$ hyperbolic Laplacian

I saw two different definitions for the weight $k$ non-Euclidean Laplacian. First, in Daniel Bump's book Automorphic Forms and Representations, the following definitions are given for smooth $\mathbb ...
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How to solve linear differential-difference equation?

Given a linear differential-difference equation: $$A_{n+2}+\partial A_{n+1}+\partial^2 A_n=0,$$ where $A$ is a function of $n$ and $x$, and $\partial$ represents the derivative about $x$. How to ...
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Definition of Differential operator

Definition 2.2, page 19 Let $M$ be a smooth manifold and $E_i \rightarrow M$ be two smooth vector bundles. A PDO $P:\Gamma (M,E_0) \rightarrow \Gamma(M,E_1)$ of order $k$ is a a linear map which ...
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Exercise about the Sturm-Liouville problems

Under what condition on the constants $c$ and $c'$ are the boundary conditions $$f(b)=cf(a)$$ and $$f'(b)=c'f'(a)$$ self-adjoint for the operator the operator $$\mathcal{L}f=\frac{d}{dx}\left(p_0(x)\...
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Formula for the application of a linear differential operator to the product of exponential and polynomial functions

In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients: $$ P\left(\frac{d}{dx}\...
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Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
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Is this differential operator Hermitian?

The operator is $$\hat{A} = -i \left(x \frac{d}{dx} + \frac{1}{2} \right).$$ Is it true that $$\langle \hat{A} \psi_1(x)|\psi_2(x)\rangle = \langle \psi_1(x)|\hat{A}\psi_2(x)\rangle\ ?$$ Here, $\...
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Generalized linear transport equation

I stumbled upon a transport equation of the form $$u_t(x,t)=u_x(x,t) + u_x(1,t).$$ Since I can write it in the form $u_t(x,t) = Lu(x,t)$ where L is some linear operator I thought that there must be ...
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Calculating Curl of a vector field using properties of $\nabla$.

So i need to find the curl of a vector field $v=(a\cdot r)a\times r$ where $a$ is some constant vector and $r=(x,y,z)$ is the position vector. So i know the curl is given by the cross product $$\...
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Elliptic boundary condition eigenvalue problem

In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem Theorem 1....
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A functional recurrence relation with differentiation, closed form

I'm interested in the way of solving the following recurrence relation: $$a_{n+1}=a_n'+a_1 a_n-b_1 b_n \\ b_{n+1}=b_n'+b_1 a_n+a_1 b_n$$ Where all $a_n,b_n$ are functions of $x$, and $'$ means ...
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Method to factorize/diagonalize differential operators with field redefinitions.

Assuming that $h_{ab}$ is some metric perturbation and $h_{<ab>}$ means the traceless part, I have the LHS of the next equation and I want to find a field redefinition $t_{ab}$ as a function of ...
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Does the formula $\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$ have a name?

Using integration by parts, we can show that: $$\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$$ where $a$ is a real number, $D$ is differentiation, and $D+a$ is the corresponding ...
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Are there good tutorial ressources to train myself on vector field equations

I've been interested for years in the formalism of maxwell equations or general relativity, but i cannot figure out how those equations including differential operators like gradient, divergence, ...
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References for Grothendieck's approach to differential calculus

What are good references for Grothendieck's approach on differential calculus? I mean here the theory alluded to in this nLab page. I could find the following: Stacks Project's chapter on Crystalline ...
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What is the relation between Grothendieck's approach to differential calculus, $\mathscr{D}$-modules, and crystalline cohomology?

It appears that the theory of regular differential operators preceded both $\mathscr{D}$-modules and crystalline cohomology (which uses, If I understood it right, a $p$-adic version of Grothendieck's ...
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Determining domain of differential operators in a concrete semigroup problem

Let s>3/2, $H^s=H^s(\mathbb{R})$ be the Sobolev space of order $s$, $B$ be the set of bounded operators from $H^{1/2}$ to itself, $u\in H^s$ and $A(u):=u\partial_x$ an operator. How can I determine ...
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Representation of ordinary differential operators in terms of a given regular operator

I'm trying to understand Lemma 3.2 from p. 355 of the paper R. C. Carlson and K. R. Goodearl, Commutants of Ordinary Differential Operators, Journal of Differenial Equations 35 (1980), 339–365. ...
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Derivative of $\frac{d}{dt} f(\gamma(t))$ with differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ a $C^1$ function (i.e. real and imaginary part $f_1, f_2$ are continuously differentiable, where $f=f_1 + i \cdot f_2$) and let $\gamma: \mathbb{R} \...
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$\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$ is invariant under all matrices

Consider the differential operator $D:=\sum_{i=1}^nx_i\frac{\partial}{\partial x_i}$. Then I claim that $D(u\circ A)(x)=D(u)(Ax)$. The proof is as follows: First note that $Du(x)=\langle x, \nabla_x u(...
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Eigenvalues of the Divergence Operator

I am researching the spectrum of eigenvalues for the divergence operator on Riemannian manifolds and how they deform tensor fields. This is mainly motivated by trying to understand dynamical systems ...
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Find Adjoint of Product & Sum of Differential Operator

I am asked to find the adjoint of the following differential operators: \begin{equation} L_{1} = a(x)\frac{\partial}{\partial x}b(x)\frac{\partial}{\partial x}, \end{equation} and \begin{equation} ...
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General formula for Differentiation Operator

I was considering the Operator $$ x\,\frac{\rm d}{{\rm d}x} $$ and applying it $n$ times to an arbitrary function $f(x)$. Is there a general formula for it? I started with the first few \begin{align} ...
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Symbol map for sheaves of differential operators

For a complex manifold $X$, its sheaf of differential operators $\mathcal{D}_X$ is a sheaf of filtered algebras, and there is an isomorphism of sheaves of graded algebras $$\text{gr } \mathcal{D}_X \...
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If differential operators are linear operators, what might it mean to act a differential operator to a function to its left?

Given a differential operator like the regular derivative, or grad or curl or div etc, it can act on a function to its right to yield a new function. Because it is linear, it is effectively like a an ...
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Thoroughly understanding the LBB condition

I'm a mechanical engineer who's just gotten into FE analysis. The more I read about FE methods for Navier-Stokes, the more I run into the "LBB condition". I understand that it talks about the order of ...
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$\frac{dx}{dt} = p, \frac{dy}{dt} = q$: Solution of these ODE imply the solution is constant along characteristics of the form $qx − py = constant$.

My lecture notes state the following: When we were dealing with first order equations we saw that a differential operator of the form, $$p\frac{\partial}{\partial{x}} + q\frac{\partial}{\...
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Differential operators on quaternions

If $\Bbb H$ denotes the quaternions one can write every element $a+bi+cj+dk\in \Bbb H$ as $1(a+bi)+j(c-di)$, so $\{1,j\}$ is a $\Bbb C$-basis of $\Bbb H$. Now let $w_1,\ldots, w_n$ be the standard ...
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How/Why does factoring the linear/differential operator suggest a specific change of variables?

Find a solution of the PDE $u_{tt} - c^2 u_{xx} = 0$, (where $c$ is a constant) in the half plane $t > 0$ with initial conditions $u(x, 0) = g_0(x)$ and $u_t(x, 0) = g_1(x)$. $$\therefore \...
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Confused about the laplace operator when used in partial differential equations.

I have a function $u(r,t)$ that satisfies $$\frac{\partial^2u}{\partial t^2} - c^2 \Delta u = 0.$$ I'm looking at a solution of a problem with this setup and it states that the above equation can be ...
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How to compute gradient of complicated scalar function ( limit and iteration)?

I have a function which gives scalar potential: $$P(c) = \lim_{n \to \infty} \frac{1}{2^n} \ln|f^{n}_c(0)|$$ where: $c$ is complex variable $f$ is the complex quadratic polynomial $$f_c(z) = z^2 ...
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1answer
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Tricky question involving finding a relation between the time derivative of an operator and a commutator of two operators

In order to get to the parts I am stuck at, I will add the examiners' solutions to each subquestion, which is needed to get to the subquestion that I am querying. The following is a bizarre question ...
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On the diffeomorphism $\mu$ in the definition of a weakly symmetric space

Selberg in some paper introduced the notion of weakly symmetric space: Definition. A weakly symmetric (Riemannian) space is a triple $(S, G, \mu)$ with $S$ is a Riemannian manifold, $G$ a locally ...
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Differentiability of $L_A(X)=AX$

Let $G$ be a matrix group and $A \in G$. Define $L_A(X)= AX : G \to G$. Show that (1) $L_A$ is differentiable. Compute its derivative. (2) $d(L_A)_I : T_I(G) \to T_A(G)$ is an isomorphism. [ ...
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On constant-coefficient linear homogenous ODE [duplicate]

I haven't done any "serious" math in a while and I wanted to get back to it via differential equations. I remember very well the different methods to solve various sorts of linear ODEs, but I can't ...
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About the closedness of $\frac d{dx}$ operator

The example section of the closed linear operator in the wikipedia page https://en.wikipedia.org/wiki/Unbounded_operator#Closed_linear_operators says that (1) "Consider the derivative operator $A =\...
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Exponential operator acting on a product of functions

An operator $\exp \left(\tau \frac{\partial}{\partial t}\right)$ when acting on a function $f(x,t)$ can be show to satisfy the following equation: $$ \exp \left(\tau \frac{\partial}{\partial t}\right)...
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eigenvalue differential operator

I stuck to this problem for half day and have no idea how to start. Can anyone please help me. Find all the eigenvalues of the following equation $(\frac{d}{dx}+x)(y(x))=\lambda$$y(x)$ I tried to ...
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When does differentiation commute with convolution with an approximate identity?

We work in $\mathbb R^n$ and denote $r$ the distance to the origin. Define the smooth approximations of the identity $$\phi_\delta = \frac1{\delta^n \int_{\mathbb R^n}\exp\left( \frac{-1}{1-r^2} \...
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Is it correct to write $\nabla\times(\nabla\times\textbf{E})$ as $\nabla\times(\nabla\times)\cdot\textbf{E}$?

Is it correct to write $\nabla\times(\nabla\times\textbf{E})$ as $\nabla\times(\nabla\times)\cdot\textbf{E}$ ? I read this kind of expression on a book, but I wonder if it is correct to write it ...
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Unitary Operator on infinitely differentiable function [closed]

If $Ψ(x)$ is an infinitely differentiable function, And the operator $\widehat{D}=\exp(ax\frac{d}{dx})$, then show that $\widehat{D}Ψ(x) = Ψ(e^a x)$
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Why de Rham complex is a complex of D-modules?

Usually, by a complex of $R$-modules, where $R$ is a ring (or sheaf of rings), one means a sequences of $R$-modules $C_n$ and $R$-homomorphisms $d_n:C_n\to C_{n-1}$ such that $d_{n-1}\circ d_{n}=0$. ...
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Green's function for $2^\text{nd}$ order PDE

Has anyone an idea how to find the Green's function of the operator $$ \widehat{{\mathcal{O}}}=-\frac{1}{R} \partial_z^2 - \partial_R \frac{1}{R} \partial_R? $$ UPDATE: For my own record (and anyone ...