# Math Tutorials

Here contains all the math tutorials for all mathematics diciplines. These are technical how-to articles that focus on teaching you a specific skill or how to solve a specific problem. From basic algebra to advanced calculus and beyond. Self study and classroom strategy. Learn new math techniques today! ### Reduction of Order For Recursions

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This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion… ### A Novel Technique of Calculating Unit Hypercube Integrals

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Introduction In this insight article, we will build all the machinery necessary to evaluate unit hypercube integrals by a novel technique. We will first… ### The Extended Riemann Hypothesis and Ramanujan’s Sum

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Riemann Hypothesis and Ramanujan's Sum ExplanationRH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All… ### A Trick to Memorizing Trig Special Angle Values Table

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In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we… ### An Introduction to Theorema Primum

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Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the… ### Python’s Sympy Module and the Cayley-Hamilton Theorem

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Two of my favorite areas of study are linear algebra and computer programming. In this article I combine these areas by using Python to confirm that a… ### Physical Applications of the “Tan Rule”

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Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,… ### Investigating Some Euler Sums

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So, why only odd powers? Mostly because the even powers were solved by Leonard Euler in the 18th century. Since the “mathematical toolbox” at that… ### Computing the Riemann Zeta Function Using Fourier Series

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Euler's amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous… ### Geometry of Side-side-angle (SSA) and Impossible Triangles

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What is the ambiguous case? In high school geometry, the idea of proofs is often first introduced to American students. A common task is to use a basic… ### How to Solve Second-Order Partial Derivatives

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Introduction A frequent concern among students is how to carry out higher order partial derivatives where a change of variables and the chain rule are… ### The Analytic Continuation of the Lerch and the Zeta Functions

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Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's… /
Introduction This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets $$C_N^n:… ### Explore the Fascinating Sums of Odd Powers of 1/n 3 Comments / The goal is to get a little bit closer to the values of the zeta function (ζ(s)) and the eta function (η(s)) for some odd values of s. This insight is… ### SOHCAHTOA: Seemingly Simple, Conceivably Complex 7 Comments / Preface My first experience with derivatives was seeing how they are obtained from the usual definition$$f'(x)=\underset{\text{$\Delta$x}\to 0}{\text{Lim}}\frac{f… ### Demystifying Parameterization and Surface Integrals

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Introduction This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration… ### Demystifying the Chain Rule in Calculus

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Introduction There are a  number of posts on PF involving a general confusion over the multi-variable chain rule.  The problem is often caused by… ### How to Make Units of Measurement Work for You

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How do we use units? You may see one of these speed limit signs, nearly every day. Even though neither of them displays units, drivers know they are implied.… ### A Journey to The Manifold SU(2): Representations

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Part 1  Representations Image source:   6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author… ### Learn Further Sums Found Through Fourier Series

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In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be… /
Important Theorems - biased, of course Implicit Function Theorem  Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1… ### The Pantheon of Derivatives – Part IV 6 Comments / Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications… ### The Pantheon of Derivatives – Part III 3 Comments / Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance,… ### The Pantheon of Derivatives – Part II 8 Comments / Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section on complex functions (The Pantheon of Derivatives - Part… ### The Pantheon of Derivatives – 5 Part Series 3 Comments / Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often… ### Trick to Solving Integrals Involving Tangent and Secant 6 Comments / This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So… ### Using the Fourier Series To Find Some Interesting Sums 4 Comments / Preliminaries If f(x) is periodic with period 2p and f’(x) exists and is finite for -π<x<π, then f can be written as a Fourier series: $f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}… ### Learn Partial Differentiation Without Tears 6 Comments / Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's… ### Learn How to Solve the Cubic Equation for Dummies 5 Comments / Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0$, even though you don't really need such a formula,… ### Omissions in Mathematics Education: Gauge Integration 31 Comments / The current (pure) mathematics curriculum at the university is well established. Most of the choices made are sensible. But still, there are some important… ### Explaining the General Brachistochrone Problem 3 Comments / Consider a problem about the curve of fastest descent in the following generalized statement. Suppose that we have a Lagrangian system$$L(x,\dot x)=\frac{1}{2}g_{ij}(x)\dot… ### Some Misconceptions on Indefinite Integrals

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Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration that… ### Mathematical Irrationality for Dummies

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On one of my restless wanderings around the Internet, I came upon a collection of proofs of the statement that the square root of 2 is an irrational… ### Learn Complex and Irrational Exponents for the Layman

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This Insight is part of our "Young Authors" series, where talented young students showcase their knowledge.  This is how both you and I learned… ### What Are Eigenvectors and Eigenvalues in Math?

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Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides… ### Introduction to Partial Fractions Decomposition

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Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum… ### How to Solve Nonhomogeneous Linear ODEs using Annihilators

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My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations… ### Solving Homogeneous Linear ODEs using Annihilators

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In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although… ### Learn About Matrix Representations of Linear Transformations

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Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered… ### Informal Introduction to Cardinal Numbers

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Cardinal numbers We will now give an informal introduction to cardinal numbers. We will later formalize this by using ordinal numbers. Informally, cardinal… ### Is There a Rigorous Proof Of 1 = 0.999…?

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Yes.First, we have not addressed what 0.999... actually means. So it's best first to describe what on earth the notation $$b_0.b_1b_2b_3...$$… ### Errors in Probability: Continuous and Discrete Distributions

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1. Classifying as discrete, continuous, or mixed These statements (or equivalents) can be found in authoritative-seeming websites:X "A random variable… ### Frequently Made Errors in Probability: Conditionals in Natural Language

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1. Turning a verbal condition into Algebra An actual thread..."A study of auto accidents has found that 40% of all fatal accidents are… ### Some Conceptual Difficulties in the Roles of Variables and Constants

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1. Variables and Constants When is a constant not a constant?  When it varies.In the standard equation ##y=ax+b##, we are used to thinking of x and… ### A Continuous, Nowhere Differentiable Function: Part 2 