# Tagged Questions

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. (Def: http://en.m.wikipedia.org/wiki/Differential_operator)

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### How to calculate the derivative of logarithm of a matrix?

Given a square matrix $M$, we know the exponential of $M$ is $$\exp(M)=\sum_{n=0}^\infty{\frac{M^n}{n!}}$$ and the logarithm is $$\log(M)=-\sum_{k=1}^\infty\frac{(I-M)^k}{k}$$ The derivative of ...
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### Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
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### Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
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### Fokker-Planck derivation. Path integral?

I am trying to understand the development of Fokker-Planck equation as is described here. Unfortunately, I cannot understand how the first equation on page 4, \begin{multline} \frac{1}{2} ...
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### Can anyone prove this equation? (Eq. with operators)

I am trying to understand the last equation from page 2 of this pdf http://physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/kap7.pdf but I am not being able to develop as here it says. Could ...
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### What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
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### Finding an matrix for an operator.

I was attempting to find a matrix for the function $x\frac{d}{dx}$ in the span of the set $\{1,x,x^2\}$ for the the standard dot product. Could someone guide me in how to do this?
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### How to find the differential with respect to the supremum norm and $L^1$ norm

I'm given a function $F:C([0,1])\rightarrow C([0,1])$, $F(f)=f^2$ (where $C([0,1])$ is given the supremum norm) and I want to find $D_g F(f)$ for any $f,g\in C([0,1])$. I find that $D_g F(f)= 2fg$, ...
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### Closure of a differential operator

Consider $A:\mathcal{D}(A)\subset L^{2}[0,1]\to L^{2}[0,1]$ given by $$A:=-\frac{d^{2}}{dx^{2}},\qquad\mathcal{D}(A):=C^{2}_{0}(0,1)$$ Now, I assume that the closure of $A$ is its extension defined ...
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### Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right)$$ ...
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### Proving the principle symbol is globally defined

I want to prove the principle symbol is globally defined as an element \begin{align} \Gamma(T^* M / \{0\}, \textrm{Hom}(\pi^*E, \pi^* F) ) \end{align} To more specify, let me explain the definition ...
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### Different solutions to the Hermite equation

The Hermite differential equation is given as such $$y'' - 2xy'+2\lambda y=0$$ writing this in strum-liouville form you get $$-(\exp(-x^2)y')'= 2\exp(-x^2)\lambda y$$ However, in order for it to ...
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### Differential Operators and Coefficients

First question on Math StackExchange here. I have been staring at this for a bit, but wasn't quite able to get the hang of it. Here it goes. We are given \begin{align} \frac{\partial}{\partial x} = ...
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### Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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### Divergence in Definition of Laplace-Beltrami Operator

I am trying to derive an explicit formula for Laplace-Beltrami operator in global Cartesian coordinates for a special case of plane curve. I have found this article, and I would like to match their ...
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### Consider a first-order operator

Consider the first-order linear differential operator L = D + 5 a) do all solutions of lx(t)=0 a subspace of F b) find these solutions c) do all solutions of lx(t) = 1 form a subspace of F I'm no ...
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### Trial Solutions to Non-homogeneous Differential Equations

I'm having trouble finding information on something in my Differential Equations & Linear Algebra class. When you're trying to find the general solution to an nth order linear non-homogeneous ...
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### What's an example of algebra where differential operators aren't generated by order 1?

For any commutative algebra $A$ over a field $k$, one can define its algebra of differential operators $\def\Diff{\operatorname{Diff}} \Diff_*(A)$, which has a filtration by order. In many cases the ...
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### Integral of the Laplace-Beltrami Operator multiplied by a function

I have the following problem: Let $\mathcal{M}$ be a $2D$-manifold in $\mathbb{R}^3$ and let $g$ denote its metric. Furthermore it is known that $\mathcal{M}$ is a closed manifold (i.e. it has no ...
### Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain
Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...