# 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!

### The Art of Integration

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

### Reduction of Order For Recursions

/
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

/
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

/
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

/
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

/
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

/
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”

/
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

/
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

/
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

/
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

/
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

/
Introduction In this brief Insight article the analytic continuations of the Lerch Transcendent and Riemann Zeta Functions are achieved via the Euler's…

/

### The Pantheon of Derivatives – Part IV

/
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

/
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

/
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

/
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

/
This little trick is used for some integration problems involving trigonometric functions is probably well-known, but I only learned it yesterday. So…

/

### Omissions in Mathematics Education: Gauge Integration

/
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

/
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

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

### Mathematical Irrationality for Dummies

/
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

/
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?

/
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

/
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

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

### Solving Homogeneous Linear ODEs using Annihilators

/
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

/
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

/
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…?

/
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

/
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

/
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

/
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

/
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…

### A Continuous, Nowhere Differentiable Function: Part 1

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