In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve. If the curve is parameterized by its arc length, the total absolute curvature can be expressed by the formula where s is the arc length parameter and κ is the curvature.This is almost the same as the formula for the total curvature, but differs in using the absolute value instead of the signed curvature. Because the total curvature of a simple closed curve in the Euclidean plane is always exactly 2π, the total absolute curvature of a simple closed curve is also always at least 2π. It is exactly 2π for a convex curve, and greater than 2π whenever the curve has any non-convexities. When a smooth simple closed curve undergoes the curve-shortening flow, its total absolute curvature decreases monotonically until the curve becomes convex, after which its total absolute curvature remains fixed at 2π until the curve collapses to a point. The total absolute curvature may also be defined for curves in three-dimensional Euclidean space. Again, it is at least 2π (this is Fenchel's theorem), but may be larger. If a space curve is surrounded by a sphere, the total absolute curvature of the sphere equals the expected value of the central projection of the curve onto a plane tangent to a random point of the sphere. According to the Fáry–Milnor theorem, every nontrivial smooth knot must have total absolute curvature greater than 4π. (Wikipedia).
Curvature and Radius of Curvature for a function of x.
This video explains how to determine curvature using short cut formula for a function of x.
From playlist Vector Valued Functions
Find the volume of a sphere given the circumference
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Curvature and Radius of Curvature for 2D Vector Function
This video explains how to determine curvature using short cut formula for a vector function in 2D.
From playlist Vector Valued Functions
Curvature for the general parabola | Differential Geometry 13 | NJ Wildberger
We now extend the discussion of curvature to a general parabola, not necessarily one of the form y=x^2. This involves first of all understanding that a parabola is defined projectively as a conic which is tangent to the line at infinity. We find the general projective 3x3 matrix for suc
From playlist Differential Geometry
The all important concept of curvature. We look at two equations for curvature and introduce the radius of curvature.
From playlist Life Science Math: Vectors
How do you find the volume of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Learn how to determine the volume of a sphere
👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo
From playlist Volume and Surface Area
Y. Tonegawa - Analysis on the mean curvature flow and the reaction-diffusion approximation (Part 1)
The course covers two separate but closely related topics. The first topic is the mean curvature flow in the framework of GMT due to Brakke. It is a flow of varifold moving by the generalized mean curvature. Starting from a quick review on the necessary tools and facts from GMT and the def
From playlist Ecole d'été 2015 - Théorie géométrique de la mesure et calcul des variations : théorie et applications
16/11/2015 - Jean-Pierre Bourguignon - General Relativity and Geometry
https://philippelefloch.files.wordpress.com/2015/11/2015-ihp-jpbourguignon.pdf Abstract. Physics and Geometry have a long history in common, but the Theory of General Relativity, and theories it triggered, have been a great source of challenges and inspiration for geometers. It started eve
From playlist 2015-T3 - Mathematical general relativity - CEB Trimester
Zero mean curvature surfaces in Euclidean and Lorentz-Minkowski....(Lecture 1) by Shoichi Fujimori
Discussion Meeting Discussion meeting on zero mean curvature surfaces (ONLINE) Organizers: C. S. Aravinda and Rukmini Dey Date: 07 July 2020 to 15 July 2020 Venue: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conduct
From playlist Discussion Meeting on Zero Mean Curvature Surfaces (Online)
F. Schulze - An introduction to weak mean curvature flow 1
It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
F. Schulze - An introduction to weak mean curvature flow 1 (version temporaire)
It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief introduction to smooth mean curvature flow and then discuss Brakke flows, their basic properties and how
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Tensor Calculus Lecture 13b: Integration - The Divergence Theorem
This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te
From playlist Introduction to Tensor Calculus
Code-It-Yourself! Retro Arcade Racing Game - Programming from Scratch (Quick and Simple C++)
And it's Go, Go, Go! This video shows how to create a simple retro style racing game in quick and simple C++. By using simple maths and rules, quite a complex game can be presented. I like the purity of this, underneath the rules are boring, but when presented as a racing game, the enjoyme
From playlist Code-It-Yourself!
Aaron Naber - The Geometry of Ricci Curvature [2012]
slides for this talk: http://www.math.stonybrook.edu/Videos/Colloquium/direct_download.php?file=PDFs/20121011-Naber.pdf Stony Brook Mathematics Colloquium Video The Geometry of Ricci Curvature Aaron Naber [MIT} October 11, 2012 http://www.math.stonybrook.edu/Videos/Colloquium/video_sl
From playlist Mathematics
Tensor Calculus Lecture 8e: The Riemann Christoffel Tensor & Gauss's Remarkable Theorem
This course will eventually continue on Patreon at http://bit.ly/PavelPatreon Textbook: http://bit.ly/ITCYTNew Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Table of Contents of http://bit.ly/ITCYTNew Rules of the Game Coordinate Systems and the Role of Te
From playlist Introduction to Tensor Calculus
A. Song - On the essential minimal volume of Einstein 4-manifolds (version temporaire)
Given a positive epsilon, a closed Einstein 4-manifold admits a natural thick-thin decomposition. I will explain how, for any delta, one can modify the Einstein metric to a bounded sectional curvature metric so that the thick part has volume linearly bounded by the Euler characteristic and
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Physics - Mechanics: Gravity (8 of 20) Determine The Density Of Earth
Visit http://ilectureonline.com for more math and science lectures! In this video I will calculate the density of the Earth.
From playlist PHYSICS 18 GRAVITY