# 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 $v^\mu = dx^\mu/ds$…