# 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! ### 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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The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta FunctionIntroduction In this brief Insight article the analytic continuations… ### A Path to Fractional Integral Representations of Some Special Functions

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0. Introduction As for the reference material I have used the text Special Functions by Askey, Andrews, and Roy which covers much of the theorems here… ### Explore the Fascinating Sums of Odd Powers of 1/n

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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… /
Preface My first experience with derivatives was seeing how they are obtained from the usual definition $$f'(x)=\underset{\text{\Delta x}\to 0}{\text{Lim}}\frac{f… ### Demystifying Parameterization and Surface Integrals / Introduction This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration… ### Demystifying the Chain Rule in Calculus / Introduction There are a number of posts on PF involving a general confusion over the multi-vairiable chain rule. The problem is often caused by… ### Make Units Work for You / How do we use units? You may see one of these speed limit signs, nearly every day. Even though neither of them display units, drivers know they are implied.… ### A Journey to The Manifold SU(2): Representations / Part 1 Representations Image source:  6. Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author… ### Further Sums Found Through Fourier Series / In an earlier insight, I looked at the Fourier series for some simple polynomials and what we could deduce from those series. There is a lot more to be… ### The Pantheon of Derivatives - Part V / Important Theorems - biased, of course Implicit Function Theorem  Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1… ### The Pantheon of Derivatives - Part IV

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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

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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

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Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section of complex functions (The Pantheon of Derivatives - Part… ### The Pantheon of Derivatives - 5 Part Series

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Differentiation in a Nutshell I want to gather the various concepts at one place, to reveal the similarities between them, as they are often… ### Trick to Solving Integrals Involving Tangent and Secant

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This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So… /
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: $f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}… ### Learn Partial Differentiation Without Tears / Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's… ### Solving the Cubic Equation for Dummies / Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0$, even though you don't really need such a formula,… ### Omissions in Mathematics Education: Gauge Integration

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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

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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… ### 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?

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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… ### Solving 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… ### 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… ### Conceptual Difficulties in the Roles of Variables

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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

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This is Part 2 of a series of articles in which the goal is to exhibit a continuous function which is nowhere differentiable, and to explore some interesting… ### A Continuous, Nowhere Differentiable Function: Part 1

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When studying calculus, we learn that every differentiable function is continuous, but a continuous function need not be differentiable at every point.…