Curvature (mathematics) | Multivariable calculus
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature. For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor. (Wikipedia).
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
6C Second equation for curvature on the blackboard
In this lecture I show you a second equation for curvature.
From playlist Life Science Math: Vectors
6D Third equation for curvature on the blackboard
In this video I introduce a third equation for curvature. Now you know them all.
From playlist Life Science Math: Vectors
Reference: Differential Geometry by Do Carmo My first video! Thank you for coming and any suggestion is very welcomed! #some2
From playlist Summer of Math Exposition 2 videos
An introduction to curvature, the radius of curvature, and how you can think about each one geometrically.
From playlist Multivariable calculus
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
Curvature of a Riemannian Manifold | Riemannian Geometry
In this lecture, we define the exponential mapping, the Riemannian curvature tensor, Ricci curvature tensor, and scalar curvature. The focus is on an intuitive explanation of the curvature tensors. The curvature tensor of a Riemannian metric is a very large stumbling block for many student
From playlist All Videos
Lecture 15: Curvature of Surfaces (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Kyle Broder -- Recent Developments Concerning the Schwarz Lemma
A lecture I gave at the Beijing International Center for Mathematical Research geometric analysis seminar. The title being Recent Developments Concerning the Schwarz Lemma with applications to the Wu--Yau Theorem. This contains some recent results concerning the Bochner technique for the G
From playlist Research Lectures
Lecture 16: Discrete Curvature I (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Lecture 17: Discrete Curvature II (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Bobo Hua (7/27/22): Curvature conditions on graphs
Abstract: We will introduce various curvature notions on graphs, including combinatorial curvature for planar graphs, Bakry-Emery curvature, and Ollivier curvature. Under curvature conditions, we prove some analytic and geometric results for graphs with nonnegative curvature. This is based
From playlist Applied Geometry for Data Sciences 2022
P. Burkhardt-Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow (vt)
We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starti
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
P. Burkhardt-Pointwise lower scalar curvature bounds for C0 metrics via regularizing Ricci flow
We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starti
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Tensor Calculus Lecture 14f: Principal Curvatures
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
Richard Hamilton | The Poincare Conjecture | 2006
The Poincare Conjecture Richard Hamilton Columbia University, New York, USA https://www.mathunion.org/icm/icm-videos/icm-2006-videos-madrid-spain/icm-madrid-videos-22082006
From playlist Number Theory
Paula Burkhardt-Guim - Lower scalar curvature bounds for $C^0$ metrics: a Ricci flow approach
We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that there is a Ricci flow definition which is stable under greater-than-second-order p
From playlist Not Only Scalar Curvature Seminar