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. …

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An Alternative Approach to Solving Collision Problems

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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…

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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…

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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 $v^\mu = dx^\mu/ds$…