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Monthly Archives: September 2011
Lie Group and Lie Algebra Representations
Given a matrix Lie group $G$, a representation $\Pi$ of $G$ is a Lie group homomorphism $\Pi: G\longrightarrow\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space and the general linear group $\mathrm{GL}(V)$ is the set of all linear isomorphisms of … Continue reading
Posted in Representation Theory
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The Lie Algebra of the Orthogonal Group $\mathrm{O}(n)\ (\mathrm{SO}(n))$
It can be easily shown that $${\rm SO}(2)=\left\{\left(\begin{array}{cc} \cos\theta & \sin\theta\\ \sin\theta & \cos\theta \end{array} \right): \theta\in[0,2\pi)\right\}\cong{\rm S}^1=\{e^{i\theta}: \theta\in[0,2\pi)\}.$$Let $\gamma(t)=\left(\begin{array}{cc} \cos\theta(t) & \sin\theta(t)\\ \sin\theta(t) & \cos\theta(t) \end{array} \right)\in\mathrm{SO}(2)$ with $\theta(0)=0$ and $\dot\theta(0)\ne 0$. Then $\gamma(t)$ be a differentiable (regular) curve … Continue reading
Posted in Lie Groups and Lie Algebras
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Matrix Lie Groups
Definition. A group $(G,\cdot,{}^{1},e)$ is a Lie group if $G$ is also a differentiable manifold and the binary operation $\cdot: G\times G\longrightarrow G$ and the unary operation (inverse) ${}^{1}: G\longrightarrow G$ are smooth maps. A subgroup of a Lie group … Continue reading
Posted in Lie Groups and Lie Algebras
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Electrostatic Potential in a Hollow Cylinder
An electrostatic field $E$ (i.e. an electric field produced only by a static charge) is a conservative field, i.e. there exists a scalar potential $\psi$ such that $E=\nabla\psi$. This is clear from Maxwell’s equations. Since there is no change of … Continue reading
Posted in Electromagnetism, Mathematical Physics
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Bessel Functions of the First Kind $J_n(x)$ II: Orthogonality
To accommodate boundary conditions for a finite interval $[0,a]$, we need to consider Bessel functions of the form $J_\nu\left(\frac{\alpha_{\nu m}}{a}\rho\right)$. For $x=\frac{\alpha_{\nu m}}{a}\rho$, Bessel’s equation (9) in here can be written as $$\rho^2\frac{d^2}{d\rho^2}J_\nu\left(\frac{\alpha_{\nu m}}{a}\rho\right)+\frac{d}{d\rho}J_\nu\left(\frac{\alpha_{\nu m}}{a}\rho\right)+\left(\frac{\alpha_{\nu m}^2\rho}{a^2}\frac{\nu^2}{\rho}\right)J_\nu\left(\frac{\alpha_{\nu m}}{a}\rho\right)=0.\ \ \ \ … Continue reading
Posted in Engineering Mathematics, Mathematical Physics
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MAT 101 Online Lecture Notes: 2.4 Analyzing Graphs of Quadratic Functions
There are two important topics in this section: graphing the quadratic function $f(x)=ax^2+bx+c$ and finding the (absolute) maximum or the minimum value of $f(x)=ax^2+bx+c$. First the sign of the leading coefficient $a$ tells us some information about the graph. If … Continue reading
Posted in College Algebra
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MAT 101 Online Lecture Notes: 2.3 Quadratic Equations, Functions and Models
Main topic in this section is solving a quadratic equation $ax^2+bx+c=0$. There are three ways to solve a quadratic equation. The first one is 1. By Factoring: This is a typical method to solve a quadratic equation whenever the polynomial … Continue reading
Posted in College Algebra
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Cylindrical Resonant Cavity
In this lecture, we discuss cylindrical resonant cavity as an example of the applications of Bessel functions. Recall that electromagnetic waves in vacuum space can be described by the following four equations, called Maxwell’s equations (in vacuum) \begin{align*} \nabla\cdot B=0,\\ … Continue reading
Posted in Electromagnetism, Mathematical Physics
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Bessel Functions of the First Kind $J_n(x)$ I: Generating Function, Recurrence Relation, Bessel’s Equation
Let us begin with the generating function $$g(x,t) = e^\frac{x}{2}\left(t\frac{1}{t}\right).$$ Expanding this function in a Laurent series, we obtain $$e^\frac{x}{2}\left(t\frac{1}{t}\right) = \sum_{n=\infty}^\infty J_n(x)t^n.$$ The coefficient of $t^n$, $J_n(x)$, is defined to be a Bessel function of the first kind of … Continue reading
Modeling a Vibrating Drumhead III
In the previous discussion, we finally obtained the solution of the vibrating drumhead problem: $$u(r,\theta,t)=\sum_{n=0}^\infty\sum_{m=1}^\infty J_n(\lambda_{nm}r)\cos(n\theta)[A_{nm}\cos(\lambda_{nm} ct)+B_{nm}\sin(\lambda_{nm}ct)].$$ In this lecture, we determine the Fourier coefficients $A_{nm}$ and $B_{nm}$ using the initial conditions $u(r,\theta,0)$ and $u_t(r,\theta,0)$. Before we go on, we … Continue reading
Posted in Partial Differential Equations
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