# 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. ### The Rise of AI in STEM - Part 2

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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

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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… ### An Alternate Approach to Solving 2 Dimensional Elastic Collisions

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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

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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. … ### An Introduction to the Generation of Mass from Energy

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Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles… ### Guide to C++ Programming For Beginners

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Contents 1. Getting a C++ Compiler and Compiling Your First Program 2. Simple Datatypes and Declarations 3. Operators and Expressions 4. Input and… ### 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… ### 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'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 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… ### 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… ### 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… ### 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… ### 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… ### 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: 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,… ### 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 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 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… ### 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… ### 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,… ### 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… ### 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… ### 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.… ### 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:  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  Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in$$U_1… ### Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity

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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

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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… ### Frames of Reference: Linear Acceleration View

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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

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Fabry-Perot Effect: The Fabry-Perot effect is usually treated in most optics textbooks as the interference that results from multiple reflections of the… /
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: $f(x)=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}… ### The Schwarzschild Geometry: Physically Reasonable? / In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres… ### The Schwarzschild Geometry: Coordinates / At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates: ds^2… ### The Schwarzschild Geometry: Key Properties / Not long after Einstein published his Field Equation, the first exact solution was found by Karl Schwarzschild. This solution is one of the… ### Learn Partial Differentiation Without Tears / Differentiation is usually taught quite well. Perhaps that's because it is the first introduction to calculus, which is considered a big step in a student's… ### Solving the Cubic Equation for Dummies / Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0$, even though you don't really need such a formula,… ### Explaining How Rolling Motion Works

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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

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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

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