### 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

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

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

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

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: [24]
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 [1]
Jacobi Matrix (Chain Rule).
Let ## (x_0,y_0 ) ## be a point in$$U_1…

### 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 of 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 at 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…

### Using the Fourier Series To Find Some Interesting Sums

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}…

### 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[/itex], even though you don't really need such a formula,…

### 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…

### 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?

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…

### Solving 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…

### 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 [tex]b_0.b_1b_2b_3...[/tex]…

### 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…

### Conceptual Difficulties in the Roles of Variables

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 which 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.…