Questions tagged [differential-operators]
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
508
questions
2
votes
1answer
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Can the del operator be “separated” like this?
In class, we were asked to prove the identity $\nabla\times(\Omega\vec{V})=\Omega(\nabla\times\vec{V})-\vec{V}\times\nabla\Omega$. One of the possible approaches involved separating the del operator ...
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0answers
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+250
Existence of a cut-off function
I have a question about the existence of a "cut-off" function with a specific purpose:
Question. Let $g:\mathbb R\to \mathbb R$ be a locally bounded function. Does there exist a sequence of ...
1
vote
0answers
13 views
Good filtrations on $A_n(K)$ modules
We are reading J. E. Björk's book: Rings of Differential Operators and we don't understand one step at Lemma 3.4:
Let $\Gamma$ and $\Omega$ be two filtrations on the left $A_n(K)$-module $M$ and ...
2
votes
1answer
51 views
Loewy decomposition of differential operators
The paper by Fritz Schwarz, "Loewy decomposition of linear differential equations", contains the following lemma, which I try to prove in order to understand the algorithm which Schwarz ...
1
vote
1answer
37 views
Differential operator; its algebraic manipulation in deriving the Laplacian ($\nabla ^2$) formula for polar coordinates
I tried out using $r = \sqrt{x^2+y^2}$ and $\tan \theta = y/x$, then getting the formula for $\partial/\partial x$ using the chain rule, and then obtaining the second partial derivative operator ...
2
votes
0answers
95 views
Explicit expression of the operator $d^*d$ on 1-forms
I tried to compute the explicit expression for $d^*d\omega$ where $\omega=\omega_jdx^j$ is a 1-form on a manifold $M^n$. My hope was to find an expression that doesn't involve the Christoffel symbols. ...
1
vote
1answer
31 views
Extending the Spectral Theorem of Unbounded Self-Adjoint Operators on Infinite-Dimensional Hilbert Spaces
I'm a physics student trying to do the maths of the Hilbert space in quantum mechanics with a bit more rigour than I'm accustomed to. I am trying to find ways to extend the spectral theorem for ...
0
votes
1answer
18 views
Inconsistent notation fro Del-Operator
It has recently come to my attention that the operations carried out with a Del operator do not match their notation.
That is, the divergence or curl are written as
$\nabla \cdot \vec{v}$ and $\nabla \...
3
votes
1answer
76 views
Why are we treating du/dx as a fraction? [duplicate]
When I was learning implicit differentiation in my class I was told to not think of $dy/dx$ as a fraction. Now I am doing integration by $u$-substitution and we treat $du/dx$ as a fraction and solve ...
0
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0answers
30 views
Determine if the given function is a linear operator
Let $\mathcal{L}(x)=a(t)x^{\prime \prime \prime}+b(t)x^{\prime \prime}+c(t)x^{\prime}+d(t)x+e(t)$ where $a(t),b(t),c(t),d(t),e(t)\text{are functions in the variable $t$}$, determine if $\mathcal{L}(x)$...
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0answers
54 views
Proof the generalization of Green's theorem
So, I was solving the problem 1.11.10 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problem was this:
Prove the generalization of Green’s ...
2
votes
0answers
95 views
Arguments to show that $\lim_{\iiint d\tau\to 0}\frac{\iint d\vec{\sigma}\times\vec{V}}{\iiint d\tau}=\nabla\times\vec{V}$
So, I was solving the problem 1.10.6 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition" and the problem was to show that:
$\lim_{\iiint d\tau\to 0}\...
5
votes
1answer
69 views
A solution for $\nabla^2\psi = k\lvert\nabla\phi\lvert^2$ with $\nabla^2\phi = 0$
So, I was resolving the problem 1.9.13 from "George B Arfken, Hans J Weber - Mathematical Methods For Physicists- Sixth edition", what a need to show is that: $\begin{equation}\psi = \frac{1}...
1
vote
0answers
23 views
Infinitesimal generator of a particular translation operator.
I have a differential operator given by
$$A u(x,t):=\sigma(t)\cdot \frac{d}{dx} u(x,t)$$
then I would like to show that this is the infinitesimal generator of a semigroup $(T_t)_{t\geq 0}$ given by
$$...
1
vote
1answer
38 views
show that $-(\vec{r} \cdot \nabla)^2\psi = -r^2\frac{\partial^2\psi}{\partial r^2} -2r\frac{\partial\psi}{\partial r}$
So, I'm trying to proof the relation $(\vec{r}\times\nabla)\cdot (\vec{r}\times\nabla)\psi = r^2\nabla^2\psi -r^2\frac{\partial^2\psi}{\partial r^2} -2r\frac{\partial\psi}{\partial r}$, where $\psi$ ...
0
votes
2answers
85 views
Hermiticity/self-adjointness of the Laplacian operator
I'm trying to understand the concept of hermicity/self-adjointness but even after reading through some threads here, I just don't get it (first problem being that I don't understand the difference ...
2
votes
0answers
61 views
Laplacian from $H^2 \to L^2$ is bounded. What is the adjoint?
Suppose I have a bounded linear operator $T: H_1 \to H_2$, where $H_1, H_2$ are Hilbert spaces. It then follows that $T^*:H_2 \to H_1$ is linear and bounded with $\lVert{T^*}\rVert = \lVert{T}\rVert$, ...
0
votes
0answers
50 views
Solution of linear differential operator
The linear differential operator
$$L=\frac{d^2}{dx^2}+2x{\frac{d}{dx}}+6 $$
is given, and then a differential equation $$Lf_1(x)=0$$ is given, where $$f_1(x)=e^{-x^2}g(x) $$ and it is assumed that $$g(...
1
vote
2answers
32 views
Problems with understanding rotor's conception
By definition rotor is:
$$ rot \vec A =
\begin{bmatrix}
(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}) \\
(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\...
0
votes
0answers
8 views
What does it mean agreement in norm mean in differential equations?
I am taking a course in numerical analysis in physical systems.
Trying to understand the next definition:
Absorbing boundary condition:
Suppose $
u$
solves the well-posed differential equation
$$
Lu=f,...
1
vote
0answers
32 views
Second order linear ODE and linear operators
Recently, I've come across this interesting but quite tricky problem.
Consider the generic second order linear ODE $a(x)\frac{d^2u}{dx^2}+b(x)\frac{du}{dx}+c(x)u(x) = \lambda r(x)u(x)$ , $\lambda \in ...
0
votes
0answers
35 views
Monotonicity of an expression involving derivatives
Let $p$ be a positive integer and $y$ be a function with class $C^p$
on some neighbourhood of $0$ in the real line. Define for any
real positive number $\kappa$
$$ H_p(\kappa) = \sum_{i=1}^p \left\{[\...
0
votes
0answers
25 views
Div, grad, rot in projective geometry?
Is it possible to define generalisations of the grad, rot and div operators
in classical projective geometry using homogenous coordinates?
I know for example in $RP^2$ that the grad of a homogeneous ...
0
votes
0answers
31 views
Show that curl$((v\cdot\nabla) v)$ = $(v \cdot \nabla)(curl(v))$
Let $v \in H^1\mathbb{(R^2)}^2$ be a divergent free vector field ($div(v)=0$). Show that $$curl((v\cdot\nabla) v) = (v \cdot \nabla)(curl(v))$$
where $H^1\mathbb{(R^2)}$ is the Sobolev Space $W^{1,2}\...
1
vote
1answer
41 views
explicit formula of $(x \frac{\partial}{\partial x})^n f(x)$
Is there any explicit expression of
$$
(x \frac{\partial}{\partial x})^n f(x)
$$
as function of $x$ and $\frac{\partial^{k}f}{\partial x^k} $$, $$ 1\leq k \leq n$.
Any idea
Thanks
0
votes
0answers
41 views
eigenfunction expansion to represent the best solution
Use the appropriate eigenfunction expansion (if it exists) to represent the best solution of the following problems.
$$u''+ u =f(x)$$ with boundary condition $$u(0) = u(2\pi), u'(0) = u'(2\pi)$$
What ...
0
votes
0answers
93 views
Center of universal enveloping algebra corresponds to bi-invariant differential operators
Let $\mathfrak{g}$ be a Lie algebra and $U(\mathfrak{g})$ be its universal enveloping algebra, that is $$U(\mathfrak{g}) = \bigotimes\mathfrak{g}/\mathcal{K},$$ where $\bigotimes\mathfrak g$ denotes ...
2
votes
1answer
70 views
Legendre operator's eigenvalues and eigenfunctions
Consider the Lengendre operator which is a Sturm-Liouville operator defined in [-1,1] give as:
$$
\begin{equation}
\mathcal{L} u=-\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d u}{d x}\right]=-\left(...
0
votes
1answer
40 views
Transform derivatives from 2D Cartesian to axisymmetric cylindrical coordinates
Consider the 1st and 2nd derivatives (differential operations) of a function $z=f(x)$ with respect to horizontal coordinate $x$ in 2D Cartesian coordinates $(x,z)$,
$$\frac{df}{dx} \quad \text{and} \...
-2
votes
1answer
24 views
Squared linear differential operator meaning
What does the linear differential operator, $Q$, defined as
$Q = (1 + \frac{1}{\alpha}\frac{d}{dt})^2$
mean exactly? That is, in the context of ODEs.
Does it mean the following?
$Q = 1 + \frac{2}{\...
0
votes
0answers
32 views
Gateaux derivative of an operator
I calculated the Gateaux derivative of $F(u):=\int_0^1 k(t,t',u(t'))\text{d}t'$ by using a taylor series of k in u:
$k(t,t',u(t'))\approx k(a,b,u(b))+\partial_uk(a,b,u(b))u(t')+\frac{1}{2}\partial^...
1
vote
0answers
58 views
Legendre differential operator
I'm trying tu calculate the eigenvalues of Legendre differential operator, that is:
$$-\frac{d}{dx}\left( (1-x^2) \frac{du}{dx}\right) = \lambda\, u \,\,\,\text{in}\,\,\,x\in[-1,1]$$
Using series of ...
0
votes
0answers
25 views
How do I apply $\nabla$ on this expression for a plane wave?
When trying to use the del function on the following equation (1) representing a linearly polarized, monochromatic, plane waves traveling in the z-direction:
$$\textbf{E} (\textbf{r},t) = \textbf{E}_0 ...
1
vote
0answers
37 views
The Laplace Operator and definition of Linear Differential Operator
I am starting Shubin's book on partial differential equations. I am just reconciling some basics. First, we define a linear differential operator like this.
$$A = \sum_{|\alpha| \le m}a_\alpha(x)D^\...
2
votes
0answers
62 views
Lower bound on the eigenvalues of a Sturm–Liouville operator
I'm considering this simplified Sturm–Liouville problem:
$$
\ddot{y} +q(x)y = -\lambda y
$$
$$
\dot y(0) = \dot y(1)= 0
$$
in the general case where $q(x)$ is sign indefinite. I'm interested on what ...
0
votes
1answer
34 views
Total differential operator - intuitive interpretation
I am having doubt regarding the total differential operator.
Let us say function Φ be a function of variables x,y,z which are dependent on t i.e.
$$
Φ=(x(t),y(t),z(t))
$$
I have read about the ...
0
votes
1answer
30 views
How to properly indicate operator notation on the Smoluchowski equation
in the case of homgeneous temperature $T$,drag coefficient $\gamma$ and conservative force $F(r)=-\nabla V(r)$, the well known Smoluchowski equation is
\begin{align}
\frac{\partial P(r,t)}{\partial t}...
-5
votes
2answers
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Inverse $D$ operators question
Find $$\dfrac 1 {D^2+6D+9}e^{-3x}$$
So I am new to the topic of $2$nd order linear ordinary differential equations and on $d$-operators. I attempted the question below and am I not supposed to just ...
0
votes
1answer
30 views
Kernel of linear differential operator is isomorphic to $\mathbb{R}^2$.
Let $I\subset\mathbb{R}$ be an open interval. Consider the following second order IVP
\begin{align*}
y''+a(x)y'+b(x)y=0,\ y(x_0)=y_0,\ y'(x_0)=y_1,\ x_0\in I,
\end{align*}
where $a,b:I\rightarrow\...
1
vote
1answer
43 views
Definition of spectrum of $\mathcal{L}$ is continuous
What does it mean for the spectrum of eigenvalues of differential operator $\mathcal{L}$ to be anywhere continuous? The textbook that I'm using doesn't give the definition of a spectrum either. This ...
0
votes
1answer
95 views
Invariant subspace of differentiation operator
Let $D \in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ be the differentiation operator, and let $U$ be an invariant subspace of $D$. Suppose there exists a $p \in U$ with deg $p = k$.
a) Show that
$\mathcal{...
11
votes
1answer
190 views
Quadratic P.S.D. differential operator that is invariant under $\textrm{SL}(2, \mathbb{R})$
Given some function $f \in L^2(\mathbb{R}^2)$, I'm interested in finding a positive semi-definite differential operator $\mathcal P: L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2)$ that is quadratic ...
0
votes
2answers
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Solve the differential equation $(D^3-3D^2+4D-2)y = (e^x +\cos x)$, where $D = \frac{\mathrm{d}}{\mathrm{d}x}$.
Solve the differential equation $$(D^3-3D^2+4D-2)y = (e^x +\cos x)\,,$$ where $D = \dfrac{\mathrm{d}}{\mathrm{d}x}$.
Here, for $e^x$, we place coefficient of $x$ in place of $D$ which is $1$ but $D=1$...
0
votes
0answers
24 views
Is this a valid manipulation for differential operators?
Full disclosure, this is a homework assignment and I don't want solutions for $x(t)$ or $y(t)$.
I have following system of equations:
$$\left[
\begin{matrix}
-2x' & y' & x & y \\
...
1
vote
0answers
21 views
Derivative operator equality
So for a while now I've been trying to understand something I've read in a paper, but can't get the prove it strictly.
Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a smooth function such that $\int f(x,\...
0
votes
0answers
49 views
What's the dimension of the image of Curl in ${\rm I\!R}^2$?
When working in ${\rm I\!R}^2$, if I apply the Curl operator to a vector field, I obtain a scalar function. Now, the value of this scalar function on a certain point is the Curl on that point. I ...
3
votes
1answer
67 views
“Differential Operator” Over Polynomial Space
Let's suppose we are given the differential operator
$T \colon \mathcal{P}_2(\mathbb{C}) \longrightarrow \mathcal{P}_3(\mathbb{C})$, over the space of quadratic polynomials with complex coefficients, ...
0
votes
2answers
33 views
Why does factorizing linear differential operators work?
When a 2nd-order differential equation is represented as something like:
$\big(D^2+(a+b)D+ab\big)(y) = \big((D+a)(D+b)\big)y$.
I don't understand how it can then by solved as:
$Du+au$ where $u = (D+b)...
0
votes
1answer
36 views
Local expressions of strictly elliptic operators on manifolds
I am studying Yosida's Functional Analysis book
and I am having a bit of trouble to understand the definitions of differential and elliptic operators on a manifold (Chap XIV, Sec 2). The relevant ...
0
votes
1answer
57 views
Fractional derivative via Continuous Functional Calculus
If we study C* algebras, at one point or another we are exposed to the idea of the Continuous functional Calculus, i.e. if $\mathcal{Y}$ is a unital C* algebra, $\xi \in \mathcal{Y}$ is normal, that ...
