Questions tagged [differential-operators]
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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23 views
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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3answers
36 views
Raising a differential operator to a power
I've been reading about the shift operator $E=e^{\frac{\mathrm{d}}{\mathrm{d}x}}$, which can be represented as
$$e^{\frac{\mathrm{d}}{\mathrm{d}x}} = I + \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1}{2!} ...
1
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1answer
34 views
How do I apply the derivative operator $dT$?
I have the following equation for the torque exerted on a turbine blade.
The paper that I am reading says, "the corresponding torque is given by..."
$$dT = ρV_2ωr^22πrdr$$
My problem is that while ...
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0answers
23 views
Factoring differential operators
I need help with the proof of this fact:
An ordinary differential operator in the variable $x$ can be factored $P = AB$ if and only if $\ker(B) \subset \ker(P)$.
The forward direction is obvious, ...
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1answer
21 views
Question About Derivative Operator's Usage
Very simple question, I didn't understand the way we are using derivative operator(dy/dx) when we want to derive multiple times such as $d^{10}f(x)/dx^{10}$ for a function such as;
$$f(x)=x^{12}-4x^{...
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0answers
54 views
What is the funciton of $dx'^2$?
Let's say same point in two co-ordinate system has the following relation from partial derivatives,
$$dx'=\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy$$
and
$$dy'=\frac{\...
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0answers
18 views
Specialized theory for differential operators structured in a matrix?
I'm wondering if there are any special considerations/ideas/treatments of equations of the following sort, where if $\boldsymbol{x} = (x_1,x_2,x_3)$:
$$\begin{bmatrix}
\partial_{x_1} & \left(\...
2
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3answers
77 views
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 ...
1
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0answers
17 views
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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1answer
23 views
$\exp(a^2\partial_x^2)f(x) = ?$
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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2answers
35 views
Exponential of the product between $x$ the derivative operator of $x$ acting in a $f(x)$
The question
I'm stuck here trying to figure out how to compute and prove, the following operator action in a function:
$\exp(\varepsilon x \partial_x) f(x) = f(x \exp(\varepsilon) )$
where $\...
0
votes
1answer
24 views
Large radius limit for a differential operator on a circle
Let $(A_r)_{r\geq0}$ be a one-parameter family of linear operators, with $A_r$ being the (weak) first derivative operator on $L^2(S_r)$, $S_r$ being the one-dimensional circle, with a multiplicative ...
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0answers
4 views
Graphing Lie transport of a function
I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in ...
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64 views
Exterior derivative of the wedge product
The book I'm reading at the moment would like to show that the following coordinate definition of the exterior derivative satisfies all the axioms assumed for the operator '$\text{d}$'. The axiom I'm ...
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1answer
44 views
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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24 views
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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0answers
18 views
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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0answers
14 views
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 ?
3
votes
0answers
92 views
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 \...
3
votes
2answers
97 views
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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1answer
24 views
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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0answers
37 views
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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1answer
50 views
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 ...
0
votes
1answer
52 views
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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0answers
61 views
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 $$\mathcal{L}f=\frac{d}{dx}\left(p_0(x)\...
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0answers
27 views
Formula for the application of a linear differential operator to the product of exponential and polynomial functions
In the context of linear differential equations, I've stumbled upon the following identity for an arbitrary pair of polynomials $P$ and $Q$ with real or complex coefficients:
$$
P\left(\frac{d}{dx}\...
2
votes
1answer
37 views
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{...
3
votes
2answers
62 views
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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votes
1answer
40 views
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 ...
0
votes
1answer
86 views
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 $$\...
2
votes
0answers
26 views
Elliptic boundary condition eigenvalue problem
In Chapter 1, section 1.5 of "The Dirac Spectrum" by Nicolas Ginoux, different elliptic boundary conditions for Dirac operators are introduced. On page 24, there is the following theorem
Theorem 1....
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1answer
47 views
A functional recurrence relation with differentiation, closed form
I'm interested in the way of solving the following recurrence relation:
$$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$$
Where all $a_n,b_n$ are functions of $x$, and $'$ means ...
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0answers
14 views
Method to factorize/diagonalize differential operators with field redefinitions.
Assuming that $h_{ab}$ is some metric perturbation and $h_{<ab>}$ means the traceless part, I have the LHS of the next equation and I want to find a field redefinition $t_{ab}$ as a function of ...
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1answer
75 views
Does the formula $\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$ have a name?
Using integration by parts, we can show that:
$$\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 ...
0
votes
0answers
18 views
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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0answers
27 views
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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votes
1answer
174 views
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.
...
2
votes
1answer
54 views
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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0answers
30 views
$\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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1answer
118 views
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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0answers
15 views
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}
...
4
votes
1answer
45 views
General formula for Differentiation Operator
I was considering the Operator
$$
x\,\frac{\rm d}{{\rm d}x}
$$
and applying it $n$ times to an arbitrary function $f(x)$.
Is there a general formula for it?
I started with the first few
\begin{align}
...
2
votes
0answers
74 views
Symbol map for sheaves of differential operators
For a complex manifold $X$, its sheaf of differential operators $\mathcal{D}_X$ is a sheaf of filtered algebras, and there is an isomorphism of sheaves of graded algebras
$$\text{gr } \mathcal{D}_X \...
3
votes
1answer
48 views
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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0answers
30 views
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 ...
4
votes
1answer
53 views
$\frac{dx}{dt} = p, \frac{dy}{dt} = q$: 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,
$$p\frac{\partial}{\partial{x}} + q\frac{\partial}{\...
3
votes
0answers
37 views
Differential operators on quaternions
If $\Bbb H$ denotes the quaternions one can write every element $a+bi+cj+dk\in \Bbb H$ as $1(a+bi)+j(c-di)$, so $\{1,j\}$ is a $\Bbb C$-basis of $\Bbb H$.
Now let $w_1,\ldots, w_n$ be the standard ...
4
votes
2answers
83 views
How/Why does factoring the linear/differential operator suggest a specific change of variables?
Find a solution of the PDE $u_{tt} - c^2 u_{xx} = 0$, (where $c$ is a constant) in the half plane $t > 0$ with initial conditions $u(x, 0) = g_0(x)$ and $u_t(x, 0) = g_1(x)$.
$$\therefore \...
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0answers
57 views
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 ...
2
votes
0answers
84 views
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 ...
