Mathematics Articles

jacobian

Jacobian (Change of Variables in Multiple Integrals)

Jacobians If $x=x(u,v)$ and $y=y(u,v)$ then the Jacobian of $x$ and $y$ with respect to $u$ and $v$ is \begin{equation} \frac{\partial (x,y)}{\partial (u,v)} =J(u,v)=\left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right|=\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial y}{\partial u}\frac{\partial x}{\partial v}. \end{equation} Example. Determine the Jacobian …

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Gram-Schmidt Process

Gram-Schmidt Process and QR Factorization

Gram-Schmidt Process The Gram–Schmidt process represents a change of basis from a basis $\mathcal{B}=(v_1, v_2, \ldots,v_m)$ of a subspace $V$ of $\mathbb{R}^n$ to an orthonormal basis $\mathcal{U}=(u_1, u_2, \ldots,u_m)$ of $V$; it is most sufficiently described in terms of the change of basis matrix $R$ from $\mathcal{B}$ to $\mathcal{U}$ via, \begin{equation} M:=\begin{bmatrix} v_1 & v_2 …

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Triple Integrals in Cylindrical and Spherical Coordinates

Triple Integrals in Cylindrical and Spherical Coordinates

Cylindrical Coordinates Each point in three dimensions is uniquely represented in cylindrical coordinates by $(r,\theta ,z)$ using $0\leq r<\infty ,$ $0\leq \theta < 2\pi ,$ and $-\infty <z<+\infty .$ The conversion formulas involving rectangular coordinates $(x,y,z)$ and cylindrical coordinates $(r,\theta ,z)$ are \begin{align} & r=\sqrt{x^2+y^2} & \tan \theta =\frac{y}{x} \\ & z=z & x=r \cos …

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Orthogonal Matrix and Orthogonal Projection Matrix

Orthogonal Matrix and Orthogonal Projection Matrix

Orthogonal Transformations and Orthogonal Matrices A linear transformation $T$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ is called an orthogonal transformation if it preserves the length of vectors: $\left|\left|T(x)\right|\right|=\left|\left|x\right|\right|$ for all $x\in \mathbb{R}^n.$ If $T(x)=Ax$ is an orthogonal transformation, we say $A$ is an orthogonal matrix. Lemma. (Orthogonal Transformation) Let $T$ be an orthogonal transformation from $\mathbb{R}^n$ to …

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Orthonormal Bases and Orthogonal Projections

Orthonormal Bases and Orthogonal Projections

Orthonormal Bases and Orthogonal Projections The norm of a vector $v$ in $\mathbb{R}^n$ is $$ \left|\left| v \right|\right| = \sqrt{v \cdot v}. $$ A vector $u$ in $\mathbb{R}^n$ is called a unit vector if $\left|\left|u\right|\right|=1.$ Example. If $v\in \mathbb{R}^n$ and $k$ is a scalar, then $\left|\left| k v\right|\right| =|k| \left|\left| v\right|\right| $, and if $v$ …

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Tonelli-Shanks Algorithm

Tonelli-Shanks Algorithm (by Example)

Solving quadratic congruence equations using a pseudo-random (Tonelli-Shanks) algorithm is discussed. We give several examples and many workable exercises. Introduction to Tonelli-Shanks Algorithm The Tonell-Shanks algorithm (sometimes called the RESSOL algorithm) is used within modular arithmetic where $a$ is a quadratic residue (mod $p$), and $p$ is an odd prime. Tonelli–Shanks cannot be used for …

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Coordinates

Coordinates (Vectors and Similar Matrices)

Coordinate Vectors Definition. Let $\mathcal{B}=(v_1,\ldots,v_n)$ be a basis of a subspace $V$ of $\mathbb{R}^n.$ For any $x \in V$ we can write $v= c_1 v_1 + \cdots +c_m v_m.$ The scalars $c_1,\ldots,c_m$ are called the $\mathcal{B}$-coordinates of $x$ and the vector $$ \left [ x \right ]_{\mathcal{B}}:= \begin{bmatrix}c_1\ \vdots \ c_m\end{bmatrix} $$ is called the …

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Quadratic Congruences

Quadratic Congruences and Quadratic Residues

First we discuss transforming and solving quadratic congruence equations. We then study quadratic residues using the Legendre symbol. Euler’s and Gauss’s Criterions are motivated and then the infamous Law of Quadratic Reciprocity is understood. General Quadratic Congruence Consider the general quadratic congruence, $$a x^2+b x+c\equiv 0 \pmod{p}$$ where $p$ is an odd prime and $(a,p)=1.$ …

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Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates

Fubini’s Theorem in Polar Coordinates The polar conversion formulas are used to convert from rectangular to polar coordinates: \begin{equation} x=r \cos \theta, \quad y=r \sin \theta, \quad r=\sqrt{x^2+y^2}, \quad \tan \theta =\frac{y}{x}. \end{equation} Theorem. (Fubini’s Theorem in Polar Coordinates) If $f$ is continuous in the polar region $R$ described by $$ 0\leq r_1(\theta )\leq r\leq …

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Euler's Totient Function and Euler's Theorem

Euler’s Totient Function and Euler’s Theorem

We prove several properties of Euler’s Totient Function and give many examples. We also discuss solving functional equations and reduced residue systems. One of Euler’s most important theorems is then demonstrated and proven. Introduction to Euler’s Totient Function Definition. For each integer $n>1,$ let $\phi (n)$ denote the number of positive integers less than $n$ …

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Alternating Series Test

Alternating Series Test (and Conditional Convergence)

Infinite series whose terms alternate in sign are called alternating series. We motivate and prove the Alternating Series Test and we also discuss absolute convergence and conditional convergence. Alternating p-series are detailed at the end. Alternating Series Infinite series whose terms alternate in sign are called alternating series. Definition. An alternating series has one of …

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Applications of Congruence

Applications of Congruence (in Number Theory)

We discuss two applications of congruence problems. How to develop a divisibility test, emphasizing theory as well as usability. We then discuss the infamous Days of the Week problem. Applications of Congruence – Divisibility Tests Our first application of congruence is a collection of theorems which help determine divisibility of an integer with another. Divisibility …

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