Geometry

ai problem solving

AI Enriched Problem Solving

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2 Comments
Introduction This article is written by way of a reference for my longstanding PF colleague and prolific poster of challenging Maths/geometry problems…
topology

The Many Faces of Topology

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Abstract Topology as a branch of mathematics is a bracket that encompasses many different parts of mathematics. It is sometimes even difficult to see…
pvsnp

Aspects Behind the Concept of Dimension in Various Fields

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Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,…
definition differences

Yardsticks to Metric Tensor Fields

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I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…
trig special functions

A Trick to Memorizing Trig Special Angle Values Table

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In calculus classes when you are asked to evaluate a trig function at a specific angle, it's 99.9% of the time at one of the so-called special angles we…
Theorema Primum tutorial

An Introduction to Theorema Primum

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Introduction Whilst no doubt most frequenters of "Physics Forums" will be familiar with Nicolaus Copernicus as the scientist who advanced the (at the…
tan rule

Physical Applications of the “Tan Rule”

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Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine,…
valentines reflections graphs

Valentine’s Reflections: Mathematical Matters of the Heart

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Introduction Being a somewhat geeky Maths 'nerd', I spent days leading up to Valentine's Day trying to find a Maths function appropriate to the day. In…
impossible triangles

Ambiguous SSA Case in Triangles — Law of Sines Explained

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What is the ambiguous case? In U.S. high schools, students are often introduced to geometric proofs for the first time. A common task is to use a small…
SOHCAHTOA

SOHCAHTOA: Seemingly Simple, Conceivably Complex

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7 Comments
What is SOHCAHTOA SOHCAHTOA is a mnemonic acronym used in trigonometry to remember the relationships between the sides and angles of right triangles.…
What is a fibre bundle

Fibre Bundles Explained: Definition, Examples & Visualization

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A fibre bundle is a mathematical structure consisting of a total space that looks locally like a simple product of two spaces but may have a different…
What is a parabola

What Is a Parabola? Equations, Focus, and Real Uses

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A parabola is a U-shaped curve where every point is equidistant from a fixed point called the focus and a fixed line called the directrix. Its standard…
What is a Tangent Line

Tangent Line Equations: Circles, Curves & Surfaces Explained

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A tangent line or tangent hyperplane to a curve or surface at a given point shares the same gradient, or first-order linear approximation, as that curve…
tensor calculus

The 10 Commandments of Index Expressions and Tensor Calculus

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16 Comments
Having more than 10 years of experience in teaching vector and tensor calculus and special and general relativity, I have noted that many people have the…
qtf_geometry

Learn the Geometry of Mathematical Quantum Field Theory

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33 Comments
This is the first chapter in a series on Mathematical Quantum Field Theory.The next chapter is 2. Spacetime. 1. Geometry The geometry of physics…

A Journey to The Manifold SU(2): Differentiation, Spheres, and Fiber Bundles

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Part 2  Differentiation, Spheres, and Fiber Bundles Image source: [24]The special unitary groups play a significant role in the standard…
deriitive4

The Pantheon of Derivatives – Lie Derivatives And Others (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…
deriitive3

The Pantheon of Derivatives – Sections, Pullbacks And Pushforwards (III)

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4 Comments
  Some Topology Whereas the terminology of vector fields, trajectories, and flows almost by itself suggests its origins and physical relevance,…
cornuspiral

Learn Frenet Equations in 2-D Which Result in the Cornu Spiral

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3 Comments
This Insights article is intended as an introduction to the Frenet-Serret equations by showing an interesting application that results from a case in two-dimensional…
geometrysimple

When Simple Geometry Unveils Deep Math

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Introduction It is a remarkable fact that consideration of very elementary concepts in geometry often leads quickly into deep and unexpected mathematical…
groupsandgeometry

Klein’s Erlangen Program: How Groups Define Geometry

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7 Comments
Felix Klein's Erlangen program defines a geometry as a set paired with a group of transformations that preserve "sameness" of figures on that set. Different…
geometry1

Pure Geometry Study Guide: Books & Roadmap for Students

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Introduction to Pure Geometry Why study pure geometry? Geometry is one of the oldest parts of mathematics. It has been studied and advanced by the greatest…
prequantumgeometry5

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

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 This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of…
prequantumgeometry4

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

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  The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement…
prequantumgeometry3

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

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  The previous article ended with the concept of classical locally variational field theories, of which a class of examples are…
prequantumgeometry2

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

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1 Comment
  The previous article motivated the importance of considering "pre-quantum field theory" in-between classical and quantum field…
prequantumgeometry

Higher Prequantum Geometry I: The Need for Prequantum Geometry

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18 Comments
 Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article,…