Differential geometry of surfaces | Surfaces | Differential geometry | Theorems in geometry | Riemannian geometry
Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting definition of Gaussian curvature makes direct use of position of the surface in space. So it is quite surprising that the result does not depend on its embedding in spite of all bending and twisting deformations undergone. In modern mathematical terminology, the theorem may be stated as follows: The Gaussian curvature of a surface is invariant under local isometry. (Wikipedia).
Linear Algebra - Lecture 33 - Eigenvectors and Eigenvalues
In this lecture, we define eigenvectors and eigenvalues of a square matrix. We also prove a couple of useful theorems related to these concepts.
From playlist Linear Algebra Lectures
Polynomials applied to an operator. Proof that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue (without using determinants!).
From playlist Linear Algebra Done Right
Ex: Find the Eigenvalues of a 3x3 Matrix
This video explains how to determine the eigenvalues of a given matrix. http://mathispower4u.com
From playlist Eigenvalues and Eigenvectors
64 - Finding eigenvalues and eigenvectors
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
Eigenvalues + eigenvectors example
Free ebook http://tinyurl.com/EngMathYT I show how to calculate the eigenvalues and eigenvectors of a matrix for those wanting to review their understanding.
From playlist Engineering Mathematics
Calculating e^A for a matrix A, explaining what this has to do with diagonalization, and solving systems of differential equations Check out my Eigenvalues playlist: https://www.youtube.com/watch?v=H-NxPABQlxI&list=PLJb1qAQIrmmC72x-amTHgG-H_5S19jOSf Subscribe to my channel: https://www.y
From playlist Eigenvalues
Symmetric matrices - eigenvalues & eigenvectors
Free ebook http://tinyurl.com/EngMathYT A basic introduction to symmetric matrices and their properties, including eigenvalues and eigenvectors. Several examples are presented to illustrate the ideas. Symmetric matrices enjoy interesting applications to quadratic forms.
From playlist Engineering Mathematics
Lecture: Eigenvalues and Eigenvectors
We introduce one of the most fundamental concepts of linear algebra: eigenvalues and eigenvectors
From playlist Beginning Scientific Computing
The Derivative of e^x is e^x Proof and Introduction to this Amazing Function
The Derivative of e^x is e^x Proof and Introduction to this Amazing Function e^x is probably the most important function in all of mathematics! In this video e start by defining e^x as the inverse of the natural logarithm of x. The graph is shown and other key things are mentioned!!! Fin
From playlist Calculus 1
Why there are no perfect maps (and why we eat pizza the way we do)
Have you ever wondered why you've never seen a perfect map? Or why bending the side of your pizza keeps the toppings from falling off? Surprisingly, these two everyday phenomena can be explained by one abstract mathematical theorem: Gauss' amazing Theorema Egregium. This video is a submi
From playlist Summer of Math Exposition 2 videos
๐ The right way to eat pizza โ approved by Gauss
A Theorem on how to Hold a Pizza Slice ๐ By folding the edge of the pizza slice in a U-shape, youโre forcing it to become flat in the direction that points towards your mouth. Why does this happen? In simple terms: A pizza slice on a plate is flat. As a consequence of a theorem by Gaus
From playlist Summer of Math Exposition Youtube Videos
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 Tensor Calculus Change of Coordinates The Tensor Desc
From playlist Introduction to Tensor Calculus
Tensor Calculus Lecture 10a: The Covariant Surface Derivative in Its Full Generality
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
An Alert for My Tensors Textbook
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
Tensor Calculus 4f: The Relationship Between the Covariant and the Contravariant Bases
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
Tensor Calculus Lecture 7c: The Levi-Civita Tensors
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
Derivative of a Basis Vector Illustrated (e_r in polar coordinates)
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
Tensor Calculus 5a: The Tensor Property
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
Tensor Calculus 4a: The Tensor Notation
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
Eigenvalues | Eigenvalues and Eigenvectors
In this video, we work through some example computations of eigenvalues of 2x2 matrices. Including a case where the eigenvalues are complex numbers. We do not discuss any intuition or definition of eigenvalues or eigenvectors, we simply carry out some elementary computations. If you liked
From playlist Linear Algebra