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!

ai problem solving

AI Enriched Problem Solving

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Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems…

Groups, The Path from a Simple Concept to Mysterious Results

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Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement…
art of integration

The Art of Integration

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Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…
recursion

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…
hypercube integral

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…
Riemann Hypothesis

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…
trig special functions

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…
Theorema Primum tutorial

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 sympy module

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…
tan rule

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,…
euler sums

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…
Fourier Series Riemann Zeta Function

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…
impossible triangles

Ambiguous SSA Case in Triangles — Law of Sines Explained

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What is the ambiguous case? In U.S. high schools, students are often introduced to geometric proofs for the first time. A common task is to use a small…
calculus

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…
lerch and zeta functions

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…
Integral Representations of Some Special Functions

The Orin Fractional Calculus

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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:…
Odd Sums

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…
SOHCAHTOA

SOHCAHTOA: Seemingly Simple, Conceivably Complex

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What is SOHCAHTOA SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…
surface integral

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…
unitsworkforyou

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.…
manifold2

A Journey to The Manifold SU(2): Representations

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Part 1 Image source: [23]  Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the…
fourierseries2

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…
deriitive5

The Pantheon of Derivatives – Important Theorems (V)

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Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||<…
deriitive4

The Pantheon of Derivatives – Lie Derivatives And Others (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…
deriitive3

The Pantheon of Derivatives – Sections, Pullbacks And Pushforwards (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,…
deriitive2

The Pantheon of Derivatives – Manifolds And Vector Fields (II)

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

The Pantheon of Derivatives – The Direction (I)

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  Differentiation in a Nutshell I want to gather the various concepts in one place, to reveal the similarities between them, as they are often…
integraltrick

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…
FourierSeries

Using the Fourier Series To Find Some Interesting Sums

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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: [itex]f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}…
partial-differentiation

Learn Partial Differentiation Without Tears

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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…
cubicfunction

Learn How to Solve the Cubic Equation for Dummies

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Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0[/itex], even though you don't really need such a formula,…
omissionguage

Why the Gauge (Henstock-Kurzweil) Integral Matters

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The gauge integral (Henstock–Kurzweil) The current (pure) mathematics curriculum at the university is well-established. Most of the choices made are…
Brachistochrone

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…
integralmisconceptions

Integration Paradoxes: Indefinite Integrals and Constants

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Introduction Integration is an incredibly useful technique taught in all calculus classes. Nevertheless, there are certain paradoxes involved with integration…
irrationalnumbers

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…
complexexponents

Understanding Exponents: Real, Irrational & Imaginary

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Introduction This is how both you and I learned exponents back in elementary school: 53 = 5 × 5 × 5 Using repeated multiplication you can raise anything…
Eigenvectors

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…
partialfractions

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…
ODE2

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…
ODE1

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…
lineartransformations

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…
cardinalnumbers

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…
999

Rigorous Proof: Why 0.999… Equals 1 (Geometric Series)

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Yes. What 0.999... Means First, we have not addressed what 0.999... means. So it is best to first describe what the notation [tex]b_0.b_1b_2b_3...[/tex]…
Distributions

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…
natural language errors

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…
variables

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 = a x + b##, we are used to thinking of…
differentiable function 2

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 that is nowhere differentiable and to explore some interesting…
differentiable function

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.…