Questions tagged [differential-operators]

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

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laplacian of a 2d tensor which only has one independent component

Could anyone guide me on how to find the following laplacian in polar coordinate? I have spent an hour searching for a reference but could not find anything helpful. T is a $2\times2$ tensor in polar ...
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Positive definite matrix and Hormander Theory

Let $\varphi \in C_0^\infty$, $\varphi \neq 0$. We'll consider the inner product in $L^2.$ Let $\alpha ,\beta$ be multi-indices, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set$$\varphi _{...
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Formal adjoint of the exterior derivative on a manifold with boundary

Let $(M,g)$ a compact Riemmanian manifold with boundary. I want to define the formal adjoint of the exterior derivative $\mathrm d\colon C^\infty(M)\to \Omega^1(M)$. If $\partial M=\emptyset$ and $\xi$...
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Invariance of irrotationality and solenoidality of a vector field under pointwise rotation

Definitions of irrotational and solenoidal vector fields are usually given for vector fields in $\mathbf{R}^2$ and $\mathbf{R}^3$, but can be generalised to vector fields in $\mathbf{R}^n$ for all $n$....
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Is $x+\frac{d}{dx}$ differential operator diagonalizable?

I would like to prove or disprove the following statement: Let $T=x+\frac{d}{dx}$ be a differential operator. Then $T$ is diagonalizable under some initial/boundary condition. Here is my attemp to ...
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Elliptic operators on bounded open sets

Let $L$ be a differential operator of order $k$ defined on a bounded open subset $U$ of a Riemannian manifold $M$ that is elliptic when restricted to a smaller open subset $V$ such that $\bar V\...
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Is it true that $(\mathbf{v}\cdot\nabla)\mathbf{v} = \mathbf{v} \cdot (\nabla \mathbf{v})$?

I have seen that sometimes the Navier-Stokes equations are written with the term $(\mathbf{v}\cdot\nabla)\mathbf{v}$ expressed as $\mathbf{v} \cdot \nabla \mathbf{v}$. However, is it true in general ...
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Finding the double derivative in the distributional sense

I want to find $u''$ in the distributional sense, where $u(x) = (1+|x|)^{-2}$ But I am very confused on how to do it. I think this is right: $\langle u'', \varphi \rangle = \langle u',\varphi ' \...
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What does $D^tf$ mean?

Let $V$ be the vector space of all polynomial functions on the real field and let $f$ be the linear functional defined by $$f(p)=\int_{0}^{1}p(x)dx.$$ If $D$ is the differential operator over $V$, ...
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Finding an operator C that satisfies AB=CA

Let $D=\frac{d}{dx}$ , $A=\sum_{i=-n}^{i=n} a_i(x)D^{i}$ and $B=b(x)D$, where $a_n(x)$ and $b(x)$ are sufficiently smooth functions and $n$ is an arbitrary positive integer. $A$ may not be invertible. ...
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A coordinate-free criterion for ellipticity of a linear differential operator

In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem. I got a bit stuck on a ...
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Please tell me the correct transformation matrix for this trig vector space

I have the transformation matrix where the transformation is the derivative operator. The basis for v is $$\{\sin(x)\cos(x), \sin^2(x), \cos2(x)\}$$ and the transformation is $$\{ \cos^2(x)-\sin^2(x),...
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Why does the d/dx operator output a derivative, whereas the integral operator accepts a differential?

Say I have differentiation operators $D_1$ and $D_2$, where $D_1$ is the normal gradient operator, and $D_2$ yields a differential $$D_2 \Big{[} y = sin(x) \Big{]}$$ $$dy = cos(x) dx$$ It would seem ...
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Proof of invariance of propagators or Green functions from their defining property rather than an explicit expression (Klein-Gordon eq.)

This same question has been asked in the physics section and answered with the usual physics proof, which is unsatisfying to me as I wish to derive the symmetry properties DIRECTLY FROM THOSE OF THE ...
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Some intuitive way to understand differential operators?

In harmonic analysis it's common to define differential operators in terms of Fourier transforms, like we often define $e^{ith(\partial_x/i)}$ as $e^{ith(\partial_x/i)}f=\mathcal F_n^{-1}(e^{ith(n)}\...
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Proof of a Douglas counterexample

Let $X$ be Banach space, $N$ a (closed) subspace of $X$ and $\pi\in X\to X/N$ the natural quotient map. Call $Y\subseteq\mathbb{Z}\to X\sqcup X/N$ the set of functions on $\mathbb{Z}$ such that $f\in ...
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Does an almost complex structure $J$ induce a differential operator on $TM$?

Let $M$ be an almost complex manifold with almost complex structure $J$. We have that $J : TM \to TM$ is a vector bundle isomorphism. Naturally, this induces a map on sections, and with abuse of ...
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How to take "limits" of operators?

Fix $h\ne0\in\mathbb{R}$. Define $\Delta_h\in\text{End}(\mathbb{R})$ by writing $\Delta_h(f):=f((-)+h)$ for all $f\in\text{Hom}_\text{Set}(\mathbb{R},\mathbb{R})$. Denote identity element of $\text{...
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galilean transformation identity

in an excericse im doing the gallilean transformation in 3D of spacetime is defined as $$ \vec{x}=\vec{x}'+\vec{v}t $$ and $$ t=t' $$ and later it says that $ \vec{\nabla}=\vec{\nabla}' $ .That I ...
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Any difference kernel acts as a differential operator

I have come across the following identity in a paper, $$\int_{-\infty}^{\infty} d\xi' \ T(\xi - \xi') g(\xi') = \hat{T}\left(-i \frac{d}{d\xi}\right) g(\xi),$$ where, $$\hat{T}(s) = \int_{-\infty}^{\...
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Calculating the Principal Part of a Differential Operator

Definition :L is quasilinear if it has the form (1) $L(u)=\sum_{|k|=n}a_k(u) \nabla^ku+f(u)$ where $a_k$ and $f$ are formal differential operators of order $\leq n-1$ As a example for a quasilinear ...
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Why is the norm of $x_n(t) = t^n$ for the differentation operator equal to 1?

I am reading Kreyszig's Introduction to Functional Analysis. He explains in an example why the differentiation operator is not a bounded operator. I cannot follow his explanation. I sometimes have ...
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PDE - Fritz John Problem 4 Chapter 2, linear operators and common solutions

I'm trying to do this problem: Let the operators $L_1, L_2$ be defined by $$L_1u=au_x+bu_y+cu ~~~; L_2u=du_x+eu_y+f_u$$ where $a,b,c,d,e,f$ are constants with $ac-bd\neq 0$. Prove that The ...
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Derivative with respect to a functional

I am coming across a derivative taken with respect to a functional and I am confused by its meaning. Say I have functions $f(x)$ and $g(x)$, say from $\mathbb{R}$ to $\mathbb{R}$, if the ...
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How to call the scalar "curl" of a vector field in a plane?

I cannot find any standard name or notation for the analog of curl of a vector field in an oriented Euclidean plane. There is this operator that to a vector field $\mathbf{F}$ given in the standard ...
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Converting a 2nd order differential operator to Hermitoan operator

I take mathematical physics course this semester and last week, my instructor introduced differential operators in class. There's something I couldn't understand. Consider that we have a 2nd order ...
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Where is the $\Vert \cdot \Vert_{\infty}$ norm Fréchet differentiable on $c_0$?

At what points $x\neq 0$ is the mapping $x\mapsto \Vert x\Vert_{\infty}$ Fréchet differentiable on $c_0:=\lbrace (x_n)_{n\in\mathbb{N}} \subset \mathbb{C}:x_n\rightarrow 0\rbrace$? A function $f$ is ...
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Definition of a tangent vector as a differential operator via Tangent Map?

Am I correct with the following reasoning: $\phantom{f}$ We defined tangent vectors on a manifold $M$ at point $p$ as equivalence classes of paths $\gamma: (-\epsilon, \epsilon) \rightarrow M$ with $\...
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Linearisation of a structured-population model with a nonlocal nonlinearity

I would like to linearise the following differential operator (which is the McKendrik-von Forster equation combined with a nonlocal nonlinearity): \begin{align} \dfrac{\partial n(t, x)}{\partial t} + \...
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Relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators

I'm very confused about the relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators. I'm following proof of a theorem in a paper, it seems that ...
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Friedrichs Inequality

I'm a little confused with the following proof of Friedrichs inequality in Lawson-Michelsohn's book “spin geometry”, page 194, Theorem 5.4: I don't understand why the last inequality $C(\|\varphi u\|_{...
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Find an ODE with given functions as solutions

I am teaching a class on differential equations lately, and in the course of trying to invent problems for a midterm, I got to wondering how to generate an ordinary differential equation having ...
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Is $\mathcal{D}_k(k[x])$ generated by $\partial_1,\partial_p,\partial_{p^2},\ldots$ and multiplication maps?

Let $k$ be a field of characteristic $p>0$, and let $\mathcal{D}=\mathcal{D}_k(k[x])$ be the ring of differential operators on $k[x]$. That is, we define $$ \mathcal{D}_{\leq -1}:=\{0\}\subseteq\...
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What is the explicit ${\rm GL}_n$-action on the Weyl algebra?

This is a typical classical invariant theory setup. Let $V = \mathbb C^n$, and let $\mathbb C[V] \cong \mathbb C[x_1,\ldots,x_n]$ be the space of polynomial functions on $V$. The group $G={\rm GL}(n,...
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Bounded operator on Sobolev spaces

I am learning about Sobolev Spaces. My professor give us the following exercise: If $a\neq b, $$\mathcal{A}=(-\infty,a)\cup(b,+\infty)$, $n,m>0$ and $\lambda$ is such that $Re(\lambda)>0$, then ...
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Does the Green's Function for an IVP always converge while integrating?

I'm having some trouble solving an ODE using the Green's function method. The problem I'm working in is the simple harmonic oscillator equation $$ L[y(t)]=f(t)$$ $$ L = \frac{d^2}{dt^2}+\omega^2$$ $$y(...
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How to prove this equality among differential operators involving chain rule?

Given the PDE $$a_{11} u_{x x}+2 a_{12} u_{x y}+a_{22} u_{y y}+a_{1} u_{x}+a_{2} u_{y}+a_{0} u=0$$ where $$\begin{bmatrix} x \\ y\\ \end{bmatrix} =\left(\begin{array}{rr} \cos\left(t\right) & -\...
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Question related to differential operator

I do not know the eg I am using can be on the site before. Actually, I was learning differential operators using a book given by our prof , and there was an example in it says like this> $\frac {d^...
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Why is operator $P = - i \frac{d}{dx} $ self-adjoint in this domain?

The domain is (where $ac$ = absolutely continuous) $$D(P) = \{f_{ac} \in L^2(\mathbb{R}) | f' \in L^2(\mathbb{R})\}$$ The text I am studying says that the functions that belong to $D(P)$ vanish at ...
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Implicit differentiation , leibniz notation- problem differential operator notation in multivariable calc

here are three examples of what im finding troublesome in understanding, i can workout the basics by myself but when it comes to iterating the differential operator i find the notation and the process ...
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How can I find an operator that returns $\frac{dx(t)}{dt}\delta(y-x(t))$ when acting on $\delta(y-x(t))$

I'm trying to find an operator that will extract the quantity $\frac{dx(t)}{dt}$ from the function $\delta(y - x(t))$. My attempt is to get something close and then subtract the part that isn't the ...
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Where the unit vectors should be on gradient operator?

I have seen two different definitions, I would say different "notations", but I would like to know which is better and more formal. The first notation (for three dimensions) is the following:...
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Free massive scalar field partition function in QFT?

I have also asked this question here. I am posting it on this forum as well in order to increase the number of people who see this question. Consider the (euclidean) path integral for the free massive ...
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Calculating criteria for Recursion Operators

I am working through Peter Olver's book Application of Lie groups to Differential Equations and I came across this example on page 311 dealing with the criteria for Recursion Operators, now the ...
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How to show skew-symmetry of the free transport operator without imposing periodic boundary conditions

I am currently dealing with Linear Kinetic Equations of the type $$ \partial_t f + v \cdot \nabla_x f = Qf $$ $Qf $ should be a bounded collision operator but is not important for this question. We ...
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Can a differential operator have infinite deficiency indices?

Suppose $T$ is a differential operator of order $m$ on $L^2(\mathbb{R})$, that is: $$ \big(Tf\big)(x) = \sum_{k \, \leq \, m} \alpha_k(x) \, f^{(k)}(x) \quad \text{ for all } \quad f \in \...
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Anything known about trace of differential operator?

Consider a self-adjoint first order linear partial differential operator $D=ia^{\mu}(x)\partial_{\mu}+a(x)$ acting on functions $\phi(x)\in L^2(\mathbb{R}^d)$. Here we have $a^{\mu *}=a^{\mu}$ and $a^*...
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Interpreting Taylor series as exponentiating an operator

In this video at 46:11, the Taylor series of function is rewritten a very different way by using exponential operator. I'll describe the method used: $$ f(x) = \sum_{i} \frac{(x-a)^i}{i!} \frac{d^i f(...
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Transformation law of elliptic operators

I am trying to understand elliptic operators and how they transform on Riemannian manifolds. It is stated in Yoshida's Functional Analysis (page 425) that if the local coordinate expression of a ...
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how to prove the following function $-F$ is the fundamental solution of the differential operator $L$

I am reading "Topics of Functional Analysis and Applications" by Kesavan by myself. This is an exercise from the first chapter. Consider the differential operator $\Big( \dfrac{d^2}{dx^2} + ...
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