Science and Math Tutorials

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


Explaining the General Brachistochrone Problem

Consider a problem about the curve of fastest descent in the…

Some Misconceptions on Indefinite Integrals

Integration is an incredibly useful technique taught in all calculus…

Basic Kinematics in Classical Mechanics

There is an interesting thing in teaching of Classical Mechanics.…

Presenting a Rare Kinematic Formula

Here we present some useful kinematic fact which is uncommon…

Elementary Construction of the Angular Velocity

Physics books  seldom contain accurate definition of the angular…

Mathematical Irrationality for Dummies

On one of my restless wanderings around the Internet, I came…

Frequently Made Errors in Vectors - Elementary Use

 A vector has magnitude and direction. Pictorially,…

LightCone7 Tutorial Part III - How Things are Computed

In Part I and Part II of this mini-series, we have briefly discussed…

Complex and Irrational Exponents for the Layman

This Insight is part of our "Young Authors" series, where talented…

LightCone7Combo Tutorial Part II - Charts

Part I dealt with the basic user interface of LightCone7Combo.…

What Are Eigenvectors and Eigenvalues?

Two important concepts in Linear Algebra are eigenvectors…

LightCone7Combo Tutorial Part I

LightCone 7 Combo is a versatile tabulating/charting cosmological…

Partial Fractions Decomposition

Partial fractions decomposition is an algebraic technique that can…

Frequently Made Errors in Climate Science - The Greenhouse Effect

1.What is meant by "The Greenhouse Effect"? Many gases, such…

Solving Nonhomogeneous Linear ODEs using Annihilators

My previous Insights article, Solving Homogeneous Linear ODEs…

Solving Homogeneous Linear ODEs using Annihilators

In this Insights article we'll look at a limited class of ordinary…

Matrix Representations of Linear Transformations

Let X and Y be finite-dimensional vector spaces. Let ##T:X\to…

Simple Python Debugging with Pdb: Part 2

This Insight article is the continuation of the first article, Simple…

Simple Python Debugging with Pdb: Part 1

I'm pretty new with Python, so I was looking around for some…

Informal Introduction to Cardinal Numbers

Cardinal numbers We will now give an informal introduction to…

Is There a Rigorous Proof Of 1 = 0.999...?

Yes.First, we have not addressed what 0.999... actually means.…

Centrifugal Force Reversal Near A Black Hole

My goal in this article is to derive a simple equation for the…

A Short Proof of Birkhoff's Theorem

Birkhoff's theorem is a very useful result in General Relativity,…
heat errors

Frequently Made Errors in Heat: Elementary Level

1. Heat, Work, Internal Energy and Kinetic Energy "If heat is…

Errors in Probability: Continuous and Discrete Distributions

1. Classifying as discrete, continuous or mixed These statements…

Frequently Made Errors: Pseudo and Resultant Forces

  1. Real versus Fictitious Pseudo, or "fictitious",…

Why Renormalisation Needs a Cutoff

This is a follow on from my paper explaining renormalisation.…
errors springs

Frequently Made Errors in Mechanics: Springs

  1. Springs in Series "A spring of constant ##k_1##…
natural language errors

Frequently Made Errors in Probability: Conditionals in Natural Language

  1. Turning a verbal condition into Algebra An actual…
impact errors

Frequently Made Errors in Mechanics: Momentum and Impacts

  An impact is an impulse (change of momentum)…

Frequently Made Errors in Mechanics: Hydrostatics

  1. Archimedes' Principle X "When a body is placed…

Conceptual Difficulties in the Roles of Variables

1. Variables and Constants When is a constant not a constant? …

A Partial "Derivation" of Gauss's Law

Gauss's law was formulated by Carl Friedrich Gauss in 1835. It…
error handling

Frequently Made Errors in Equation Handling

  1. Algebra versus Arithmetic When numerical values…
kinematic errors

Frequently Made Errors in Mechanics: Kinematics

  Kinematics is the subset of dynamics…
mistakes moments

Frequently Made Errors in Mechanics: Moments

 The term "moment" is used in various ways in Physics…

Renormalisation Made Easy

What Is The Issue With RenormalisationIf you have an interest…
electrical wire

Misconceiving Mutual Inductance Coefficients

 A commonly used formula for mutual inductance M between…

Frequently Made Errors in Mechanics - Friction

  1. Direction of the normal Definition: The normal…
differentiable function 2

A Continuous, Nowhere Differentiable Function: Part 2

This is Part 2 of a series of articles in which the goal is to…
forces mistakes

Frequently Made Errors in Mechanics: Forces

 Notation: In this page, a circumflex signifies…
2D Animation

Visualizing the 2-D Particle in a Box

The particle in a box is a staple of entry-level Quantum Mechanics classes…