Mathematics Articles 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 … 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 … 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 … 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 … Triple Integrals (With Fubini’s Theorem)

The Definition of a Triple Integral Suppose $f(x,y,z)$ is defined on a closed bounded solid region $R,$ which in turn is contained in a box $B$ in space. We partition $B$ into a finite number of smaller boxes, call this partition $P,$ we choose a representative point $\left(x_k^*, y_k^*, z_k^* \right)$ from each subdivision in … Taylor Polynomials and Approximations

Recall a tangent line approximation of a function is used to obtain a local linear approximation of the function near the point of tangency. We consider how to improve on the accuracy of tangent linear approximations by using higher-order polynomials as approximating functions. We also discuss the error associated with such approximations. Polynomial Approximations The … 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 $v$ … 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 … Surface Area Using Double Integrals

Introduction to Surface Area We apply double integrals to the problem of computing the surface area over a region. We demonstrate a formula that is analogous to the formula for finding the arc length of a one variable function and detail how to evaluate a double integral to compute the surface area of the graph … Taylor Series (and Maclaurin Series)

The Taylor series of a function is a representation as power series whose terms are calculated from the values of the function’s derivatives at a single point (the center). If the Taylor series is centered at zero, then that series is also called a Maclaurin series. We discuss the Maclaurin series of the sine and … 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 … 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 $(a,p)=1.$ … 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 $$0\leq r_1(\theta )\leq r\leq … Power Series (and Their Convergence) We introduce power series and discuss convergence of power series. Finding the interval of convergence and finding the radius of convergence is explained through several examples. We also discuss term-by-term differentiation and integration power series. What are Power Series? We begin with the definition of power series. Definition. Let x be a variable. An infinite … Linear Transformation Matrix and Invertibility The Matrix of a Linear Map Let V and W be finite-dimensional linear spaces. Definition. (The Matrix of a Linear Map) Let T\in \mathcal{L}(V,W) and let b_1={v_1,\ldots,v_n} be a basis for V and b_2={w_1,\ldots,w_m} be a base for W. Then the matrix of T with respect to the bases b_1 and b_2 is$$ \begin{bmatrix} … 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 $n$ … Fubini’s Theorem for Double Integrals

Iterate Integrals Over Rectangular Regions Green’s Theorem is named after the mathematician Guido Fubini. Theorem. (Fubini’s Theorem for Rectangular Regions) If $f$ is a continuous function of $x$ and $y$ over the rectangle $R:$ $a\leq x\leq b,$ $c\leq y\leq d,$ then the double integral of $f$ over $R$ may be evaluated by either iterated … The Ratio Test and the Root Test

The Ratio and Root Tests are criterions for the convergence of an infinite series. We provide several examples using these convergence tests and several exercises. The Ratio Test The Ratio Test is a criterion for the convergence (a convergence test) of an infinite series $\sum a_n$. It depends on the quantity \begin{equation} \label{rt} \lim_{n\to+\infty} \frac{a_{n+1}}{a_n}. … Transformation Definition and Rank-Nullity Theorem

Introduction to Linear Maps We begin with the definition of a linear transformation. Definition. Let $V$ and $W$ be linear spaces. A function $T$ from $V$ to $W$ is called a linear map if $$T(f+g)=T(f)+T(g) \qquad \text{and}\qquad T(k f)=k T(f)$$ for all elements $f$ and $g$ of $V$ and for all scalars $k.$ The … Fermat’s Theorem (and Wilson’s Theorem)

Wilson’s theorem, its converse, and Fermat’s theorem are discussed. We motivate each proof through example and careful write out the proof of each theorem. Several examples of their use are given. Wilson’s Theorem This theorem is named after one of Edward Waring‘s students, John Wilson. But actually, Wilson only observed the result to be true … Double Integrals and the Volume Under a Surface

The Volume Under a Surface Consider the rectangle given by $$R=[a,b]\times[c,d]=\{(x,y)\mid a\leq x\leq b, c\leq x\leq d\}.$$ We wish to construct a (regular) partition of $R.$ To do so, let $$a=x_0 < x_1 < x_2 < \cdots < x_{i-1} < x_i < \cdots < x_n=b$$ be a partition of $[a,b]$ into subintervals … 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 … Subspaces and Linear Independence

Definition. A subset $U$ of a vector space $\mathbb{V}$ is called a subspace of $\mathbb{V}$ if it has the following three properties (1) $U$ contains the zero vector in $\mathbb{V}$, (1) $U$ is closed under addition: if ${u}$ and ${v}$ are in $U$ then so is ${u}+{v}$, and (3) $U$ is closed under scalar multiplication: … 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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