The differential-operators tag has no wiki summary.
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Solving tensor Identities
For my homework I am required to prove the following identity. Here $\bf{u}$ is a vector.
$\nabla \bf{uu} = u(\nabla.u) + (u.\nabla)u$
Only thing I understand in this equality is lest hand side is ...
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1answer
37 views
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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1answer
22 views
Handling matrix of differential operator when using the Ritz method for an extremum problem
The internal energy (or strain energy) of a typical structural member under a displacement field $\{u\}$ can be represented as:
$$
U = \frac{1}{2}\int_{V}[u]^T[B]^T[F]^T[B][u]dV
$$
where $V$ is the ...
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1answer
51 views
Prove that D (the differential operator) maps V (a vector space) into V.
I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space.
Here's some of the ...
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2answers
73 views
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 $$\left| ...
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39 views
Doubt about the spectrum of an operator
I consider the Laplacian operator
$$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
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55 views
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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1answer
23 views
Homogenous Polynomial Functions and the Symbol of a Differential Operator
I have a trivial question concerning Lawson/Michelsohn's "Spin Geometry", Chapter III.§1. There, the symbol of a differential operator $P$ is defined to be a section $\sigma(P)$ in the bundle ...
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1answer
71 views
Justification behind changing coordinates of a differential operator
On many websites focused on physics, (say http://skisickness.com/2009/11/20/ ) they like to represent differential operators in different coordinates. I.e. going from the standard basis to polar ...
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1answer
99 views
Partial derivative of an integral operator functional
Suppose $f(x) \equiv f_0(x) + \epsilon t(x)$, where $x,\epsilon \in{\mathbb{R}}$.
And let $\mathcal{L}[f(x)] \equiv \int_a^b f(x')dx'$.
I want to differentiate $\mathcal{L}$ w.r.t. $\epsilon$. So I ...
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2answers
70 views
What does the symbol $\Delta$ stands for?
While studying Landau-Lifshitz equation following term appears,
$-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$
In above equation m is a vector quantity. It will be great if someone can ...
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47 views
Inverse of a certain differential operator (resolvent)
I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator $D$ below, and identity operator $I$,
$$
D=\begin{pmatrix}
...
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1answer
210 views
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, ...
4
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1answer
89 views
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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2answers
61 views
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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1answer
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Differential operators: elliptic vs strongly elliptic
This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic?
After ...
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2answers
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Computing $e^{isD}$ for a differential operator D
I'm trying to understand functional calculus of unbounded operators and everywhere I see proofs of its existence, but it seems that no one ever dares to compute some easy example.
Lets take $D = ...
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2answers
105 views
Transpose of the differential operator
Is the differential operator ${d\over dx}$ antisymmetric? If so, what does it even mean to take it's transpose? Thank you.
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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 ...
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0answers
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How to prove this equivalence?
Consider the general elliptic operator
$$M=\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j},$$
where $a_{ij}$ are continous functions. The function $u$ satisfies
$$|Mu|\leq A(|\nabla ...
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1answer
55 views
Laplace-like operator
Help me please to apply a Laplace-like operator:$ \Delta f:= \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2
f}{\partial z^2}
+ {1\over r}\,\frac{\partial f}{\partial r} - {f\over
r^2} $ on the ...
2
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1answer
90 views
Error in proof of self-adjointness of 1D Laplacian
I have successfully checked self-adjointness of simple and classic differential operator - 1D Laplacian
$$D = \frac {d^2}{dx^2}: L_2(0,\infty) \rightarrow L_2(0,\infty)$$
defined on
$$\{f(x) | f'' ...
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65 views
diagonalizing a simple matrix of differential operators
I have the following endomorphism (whose determinant I would like to calculate) of the (infinite dimensional) vector space $\Omega^0(M) \oplus \Omega^1(M)$ for some closed Riemannian manifold $M$:
...
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1answer
122 views
Differentiation operator applying on matrix
I need to apply a differential operator (nabla) on a matrix. Problem is, that I don't know how to calculate that. Do I treat nabla as a column vector and simply multiply vector with the matrix? Or is ...
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2answers
190 views
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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votes
2answers
77 views
Am I allowed to move around an operator like this?
Can I take this product:
$$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$
And factor out one of the $L$'s to get:
$$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$
Where the operator $\frac{d}{dt}$ now ...
2
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2answers
256 views
The minus Laplacian operator is positive definite
In a textbook of functional analysis I found this equation derived from Green's first identity
$$\int _{ \Omega }^{ }{ u{ \nabla }^{ 2 }ud\tau } =\int _{ \partial \Omega }^{ }{ u\frac { ...
0
votes
1answer
40 views
Infinite propagation speed for the Schrodinger operator
Question related to:
On the propagation of singularities in PDE and
Hypoellipticity and singular support.
in what sense is to interpret the sentence the schrodinger operator
has infinite propagation ...
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0answers
33 views
Hypoellipticity and singular support
There is a theorem that states that if $p(D)$ is a linear
partial differential operator with constant coefficients
and its fundamental solution $E$ is $C^{\infty}$ outside $\{0\}$ then
the operator ...
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2answers
186 views
Fundamental solution of the wave operator
What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks.
The operator I'm talking about is
...
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1answer
83 views
On the propagation of singularities in PDE
This question might be a little generic, but i wanted to get some idea on the concept of propagation of singularities in PDE. Searching the internet i only found very complicated things about the ...
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0answers
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Non-hypoelliptic operator
Consider the following operator $$\partial_t^2-\partial_x^2+\lambda(x,t)$$ where $\lambda$ is a $C^{\infty}$ function on $\mathbb{R}^2$ and the operators above are second partial derivatives. How can ...
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$\left(D^2-\frac{z^2}{a^2}\right)U=-\frac{z}{a}g(x)$ operator solution
The following equation arises as an intermediate step in solving a PDE. $U$ is a function of both $z$ and $x$ but for the purses of this argument $z$ is deemed constant.
$D$ denotes $\frac{d}{dx}$, ...
5
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1answer
129 views
How to prove (0,1) form is not $\overline\partial$-exact
On a complex manifold, if we are dealing with the $d$ operator, there's a pretty easy way of showing some form is not $d$-exact, simply by integrating in a closed loop. If you can find a loop that is ...
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0answers
51 views
Cayley Transform (PDE)
I really need your help in solving the following problem:
Prove that a unitary operator $U$ acting on a Hilbert space $H$ is the Cayley transform of some self-adjoint operator if and only if $1$ is ...
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Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?
Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space
$$
C=\{u\in C^2([0,1]):u(0)=u(1)=0\}.
$$
It's not hard to find the eigenvalues and ...
2
votes
1answer
78 views
How do you calculate an exterior derivative on forms in $\mathbb{R}^3$?
If we have a form, say, $\omega = f(x,y,z) \, dx + g(x,y,z) \, dy + h(x,y,z) \, dz$, what is the formula for the exterior derivative $d \omega$?
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1answer
118 views
what is difference between the square of an operator and the expectation value of that operator
operator $\hat A$ is a mathematical rule that when applied to a ket $\hat A|\phi\rangle$ transforms it into another ket $\hat A|\phi '\rangle $ and too for bra.
$\langle \phi| \hat A|\phi\rangle$ ...
3
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1answer
125 views
Subset of differential operators is a finitely generated module?
I was reading about differential operators, and there is a small claim I don't understand.
First, let $A$ be a commutative algebra over $k$, a field. We have the recursive definition for the algebra ...
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4answers
275 views
Determining the action of the operator $D\left(z, \frac d{dz}\right)$
This question was motivated by a question by Tobias Kienzler and its wonderful answers.
I begin as in the linked question...
Using the Taylor expansion
$$f(z+a) = \sum_{k=0}^\infty ...
2
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1answer
2k views
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} + ...
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1answer
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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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1answer
267 views
Symbol of a (non linear) differential operator
I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear.
In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by ...
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1answer
337 views
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 ...
4
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1answer
115 views
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 ...
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1answer
391 views
Laplacian Operator Represented as a Matrix - Problem Finding the Hermiticity
I'm trying to discretize the Laplacian operator, and represent it with a matrix, but I'm running into a problem: my result is not hermitian when it should be. Here are my calculations:
In one ...
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2answers
122 views
What does ad$f$ mean, for $f$ a smooth function?
I am currently reading Nicole Berline "Heat Kernels and Dirac Operators". On page 64 Differential Operators are introduced that are generalized from operators acting on scalar functions to vector ...
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1answer
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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., ...
