Science and Math Tutorials

Here contain the expert tutorials for all science and math disciplines.  This is a master list. Tutorials are technical and focus on one narrow skill. It’s more like a how-to than a guide.

artificial intelligence

The Rise of AI in STEM - Part 2

We asked our PF Advisors “How do you see the rise in A.I. affecting STEM in the lab, classroom, industry and or in everyday society?”. We got so many…

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…
lerch and zeta functions

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…
Integral Representations of Some Special Functions

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…
elastic ball collision

An Alternate Approach to Solving 2 Dimensional Elastic Collisions

Introduction This article follows on from the previous on an alternate approach to solving collision problems. In that article we determined the equal…

How to Recognize Split Electric Fields

Introduction In a previous Insight, A New Interpretation of Dr. Walter Lewin’s Paradox, I introduced the fact that there are two kinds of E fields. …
Mass Generation

An Introduction to the Generation of Mass from Energy

Introduction This article is essentially an addition to the previous one on (mainly) inelastic collisions to include the particular case of inelastic…
Collision problems

An Alternative Approach to Solving Collision Problems

Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles…
c++ guide for beginners

Guide to C++ Programming For Beginners

Contents 1. Getting a C++ Compiler and Compiling Your First Program 2. Simple Datatypes and Declarations 3. Operators and Expressions 4. Input and…
Odd Sums

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…
recursion in programming

Recursion in Programming and When to Use or Not to Use It

Recursion is actually quite simple. It's a subroutine calling itself. Its surprising but some problems that look quite hard can be trivial using recursion…
maxwell magneto

Maxwell's Equations in Magnetostatics and Solving with the Curl Operator

Introduction: Maxwell's equation in differential form ## \nabla \times \vec{B}=\mu_o \vec{J}_{total}+\mu_o \epsilon_o \dot{\vec{E}}  ##  with ## \dot{\vec{E}}=0…
Learn Dimensional Analysis

Learn the Basics of Dimensional Analysis

As a university teacher and as a PF member, I have often noted that students are largely unaware of or not using dimensional analysis to help them in their…
Quantum Mechanical Commutator

The Classical Limit of Quantum Mechanical Commutator

The Classical Limit of Commutator (without fancy mathematics) Quantum mechanics occupies a very unusual place among physical theories: It contains classical…
Minkowski Spacetime

Precession in Special and General Relativity

The Absolute Derivative In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which…
isotropy definition

A Formal Definition of Large-Scale Isotropy

This Insight is part of my attempt to develop a formal definition of 'large-scale isotropy', a concept that is fundamental to most cosmology, but that…
surface integral

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…
Kerr Spacetime

Fermi-Walker Transport in Kerr Spacetime

In the last two posts in this series, we developed some tools for looking at Fermi-Walker transport in Minkowski spacetime and then applied them in Schwarzschild…

Fermi-Walker Transport in Schwarzschild Spacetime

In the first post in this series, we introduced the concepts of frame field, Fermi-Walker transport, and the "Fermi derivative" of a frame field, and developed…
AVX-512 Programming subtotals

AVX-512 Programming: Extracting Column Subtotals from a Table

In this Insights article I'll present an example that shows how Intel® AVX-512 instructions can be used to read a whole row of data in a single operation,…
Minkowski Spacetime

Fermi-Walker Transport in Minkowski Spacetime

This is the first of several posts that will develop some mathematical machinery for studying Fermi-Walker transport. In this first post, we focus on Minkowski…
AVX-512 conclusion

AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic Conclusion

In the first part of this article (AVX-512 Assembly Programing - Opmask Registers for Conditional Arithmetic), we looked at how opmask registers can be…
AVX-512 registers

AVX-512 Assembly Programming: Opmask Registers for Conditional Arithmetic

This is the second installment in a continuing series of articles on Intel AVX-512 assembly programming. The first installment is An Intro to AVX-512 Assembly…
AVX-512 Assembly Programming

An Intro to AVX-512 Assembly Programming

History In 1998, the Intel Corporation released processors that supported SIMD (single instruction, multiple data) instructions, enabling processors to…
Data Structures Programming

Intro to Data Structures for Programming

IntroductionIn the first part of this series, I talked about some fundamental notions in the world of algorithms. Beyond the definition of an algorithm,…
rotational mechanics

An Example of Servo-Constraints in Mechanics

Servo-constraint was invented by Henri Beghin in his PhD thesis in 1922. For details see the celebrated monograph in rational mechanics by Paul Appell.To…
unity orbital mechanics

Orbital Mechanics in Unity Game Engine for Augmented Reality

In this post I’ll go over implementing basic orbital mechanics simulations in the Unity game engine as well as an approach to scaling the simulation…
walter lewin

A New Interpretation of Dr. Walter Lewin's Paradox

Much has lately been said regarding this paradox which first appeared in one of W. Lewin's MIT lecture series on ##{YouTube}^{(1)}##.  This lecture was…

Calculating the Spin of Black Hole Sagittarius A*

This Insight takes a look at how it is possible to calculate the spin of Sagittarius A*, the supermassive black hole at the centre of the Milky Way using…

Intro to Algorithms for Programming

Many threads here at PF include some question about how to learn programming. This is asked by Physics students who want to learn programming in order…

Solving Einstein's Field Equations in Maxima

A few months ago, pervect pointed me to a post by Chris Hillman which is an introduction to the usage of Maxima for General Relativity. Maxima is a free…

Rindler Motion in Special Relativity: Rindler Coordinates

Our destination In our last article, Hyperbolic Trajectories, we derived some facts about the trajectory of a rocket that is undergoing constant (proper)…

Rindler Motion in Special Relativity: Hyperbolic Trajectories

Introduction: Why Rindler Motion? When students learn relativity, it's usually taught using inertial (constant velocity) motion. There are lots of reasons…

Statistical Mechanics: Equilibrium Systems

This is the first of a multi-part series of articles intended to give a concise overview of statistical mechanics and some of its applications. These articles…

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.…
Maxwell equations

The Homopolar Generator: An Analytical Example

Introduction It is surprising that the homopolar generator, invented in one of Faraday's ingenious experiments in 1831, still seems to create confusion…

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…

An Integral Result from Parseval's Theorem

Introduction: In this Insight article,  Parseval's theorem will be applied to a sinusoidal signal that lasts a finite period of time.  It will be shown…

Introduction to Digital Processing

In the early days of the personal computer revolution, computers were small, simple and easy to operate. It was always great fun to write BASIC games on…

Relativity Using the Bondi K-calculus

Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity…

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…

Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity

Traditional presentations of special relativity place emphasis on "velocity", which of course has an important physical interpretation... carried over…

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…

Frames of Reference: Linear Acceleration View

My previous Insight, Frames of Reference: A Skateboarder's View, explored mechanical energy conservation as seen from an inertial frame moving relative…

Fabry-Perot and Michelson Interferometry: A Fundamental Approach

Fabry-Perot Effect: The Fabry-Perot effect is usually treated in most optics textbooks as the interference that results from multiple reflections of the…

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}…

The Schwarzschild Geometry: Physically Reasonable?

 In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres…

The Schwarzschild Geometry: Coordinates

 At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates:$$ ds^2…

The Schwarzschild Geometry: Key Properties

 Not long after Einstein published his Field Equation, the first exact solution was found by Karl Schwarzschild. This solution is one of the…

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,…

Explaining How Rolling Motion Works

Although rolling wheels are everywhere, when most people are asked "what is the axis of rotation of a wheel that rolls without slipping?", they will answer…

Grandpa Chets Entropy Recipe - Determining the Change in Entropy

How do you determine the change in entropy for a closed system that is subjected to an irreversible process?Here are some typical questions we get…

Orbital Precession in the Schwarzschild and Kerr Metrics

The Schwarzschild Metric A Lagrangian that can be used to describe geodesics is [itex]F = g_{\mu\nu}v^\mu v^\mu[/itex], where [itex]v^\mu = dx^\mu/ds[/itex]…