In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface (such as the sphere or the cylinder) at a point can be rolled around the surface to obtain the tangent plane at other points.
Properties
The tangential contact between the surfaces being rolled over one another provides a relation between points on the two surfaces. If this relation is (perhaps only in a local sense) a bijection between the surfaces, then the two surfaces are said to be developable on each other or developments of each other. Differently put, the correspondence provides an isometry, locally, between the two surfaces.
In particular, if one of the surfaces is a plane, then the other is called a developable surface: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.
Flat connections
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection on the surface (it provides an example of parallel transport along a curve), and a developable surface is one for which this connection is flat.
Development economics is a branch of economics which deals with economic aspects of the development process in low-income countries. Its focus is not only on methods of promoting economic development, economic growth and structural change but also on improving the potential for the mass of the population, for example, through health and education and workplace conditions, whether through public or private channels.
Development economics involves the creation of theories and methods that aid in the determination of policies and practices and can be implemented at either the domestic or international level. This may involve restructuring market incentives or using mathematical methods such as inter-temporal optimization for project analysis, or it may involve a mixture of quantitative and qualitative methods.
Unlike in many other fields of economics, approaches in development economics may incorporate social and political factors to devise particular plans. Also unlike many other fields of economics, there is no consensus on what students should know. Different approaches may consider the factors that contribute to economic convergence or non-convergence across households, regions, and countries.
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula
where dy/dx denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal.
There are several approaches for making the notion of differentials mathematically precise.
Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
If a groupG acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.
Differential Geometry Part 3 Transformations and the Metric Tensor
Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is easily generalilzed to higher dimensions. Some notes on vector notation and the Einstein summation convention is introduced.
published: 25 May 2009
Differential geometry lecture | What is differential geometry | History of differential geometry
#differentialgeometrylecture
#whatisdifferentialgeometry
#historyofdifferentialgeometry
This is an introduction to differential geometry. What is differential geometry? This is a very common question. I have discussed in details about differential geometry along with the history of differential geometry.
00:00 - 00:51 - Introduction
00:51- 02:24 - What is differential geometry:
02:24 - 04:53 - Branches of differential geometry
04:53 - 05:50 - The objective of this program
05:50 - 07:50 - What is a smooth manifold:
07:51 - 09:07 - What is smoothness?
09:09 - 11:08 - What do we need abstractness?
11:09 - 12:59 - Which one should I study first: Topology or differential geometry
13:00 - 13:58 - Why do we apply calculus to differential geometry
13:59 - 19:43 - Why should I learn different...
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
published: 04 Feb 2021
What is Differential geometry?, Explain Differential geometry, Define Differential geometry
#Differentialgeometry #audioversity
~~~ Differential geometry ~~~
Title: What is Differential geometry?, Explain Differential geometry, Define Differential geometry
Created on: 2018-11-10
Source Link: https://en.wikipedia.org/wiki/Differential_geometry
------
Description: Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differ...
published: 10 Nov 2018
Differential Geometry | Math History | NJ Wildberger
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a...
published: 07 May 2012
History of differential geometry | Differential geometry lecture | Modern differential geometry
#differentialgeometrylecture
#whatisdifferentialgeometry
#historyofdifferentialgeometry
This video is an extract of my original video 'What Is Differential Geometry'. As the length of the previous video is more than an hour, I extracted the 'historical' part and emphasized on that. This video gives you a short but clear idea of how the first seed of differential geometry was sown, the further developments, who were the people who contributed in this field, hyperbolic geometry, and an account on modern differential geometry.
In case you are interested to look into the complete video, here is the link:
https://www.youtube.com/watch?v=XTUe8zxgkpY&t=878s
00:00 - 01:20 - Introduction
01:21 - 10:38 - History of differential geometry
10:39 - 16:20 - Hyperbolic geometry
16:21 - 28:09 - Modern d...
published: 23 Jan 2022
Gilbert Strang: Linear Algebra vs Calculus
Full episode with Gilbert Strang (Nov 2019): https://www.youtube.com/watch?v=lEZPfmGCEk0
New clips channel (Lex Clips): https://www.youtube.com/lexclips
Once it reaches 20,000 subscribers, I'll start posting the clips there instead.
(more links below)
For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
Clip from full episode: https://www.youtube.com/watch?v=lEZPfmGCEk0 If you enjoy these clips, subscribe to the new clips channel (Lex Clips): https://www.youtube.com/lexclips Once it reaches 20,000 subscribers, I'll start posting the clips there instead. For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
(more links belo...
published: 27 Nov 2019
Feynman-"what differs physics from mathematics"
A simple explanation of physics vs mathematics by RICHARD FEYNMAN
published: 15 Oct 2018
How to learn differential geometry | Differential geometry lecture | Differential geometry msc maths
#howtolearndifferentialgeometry
#differentialgeometrylecture
#differentialgeometrymscmaths
How to learn differential geometry? In this differential geometry lecture, I have discussed about curves and surfaces. I have reviewed the best book to learn curves and surfaces which is the foundation of starting to learn differential geometry. I have also discussed about the importance of learning curves and surfaces and why you should learn curves and surfaces. This is the best and the easiest book to start learning differential geometry and provides an easy explanation to the problems with necessary solutions.
00:00 - 01:22 - Introduction
01:23 - 03:36 - Why you need to learn curves
03:37 - 07:13 - Importance of curves in differential geometry
07:14 - 09:14 - Curves are the foundation of differ...
published: 25 Jun 2023
Differential geometry msc mathematics | Differential geometry | How to learn differential geometry
#differentialgeometrymscmathematics
#differentialgeometry
#howtolearndifferentialgeometry
How to learn differential geometry? What are the best books for differential geometry? What is the sequence of study? What are the mathematical pre-requisites? All these questions are answered in this video. In this video, I have explained about the books and the method of studying, I have given a step by step approach, first starting with building the foundation, concepts, books that you should read and then finally moving on the text books of differential geometry. I have shown the content and also explained the mathematics that you need to know along with the assumptions that the book makes.
00:00 - 01:33 - Objective of this video
01:34 - 02:20 - Topics
02:21 - 03:16 - A quick recap
03:17 - 03:57...
Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is e...
Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is easily generalilzed to higher dimensions. Some notes on vector notation and the Einstein summation convention is introduced.
Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is easily generalilzed to higher dimensions. Some notes on vector notation and the Einstein summation convention is introduced.
#differentialgeometrylecture
#whatisdifferentialgeometry
#historyofdifferentialgeometry
This is an introduction to differential geometry. What is differential ...
#Differentialgeometry #audioversity
~~~ Differential geometry ~~~
Title: What is Differential geometry?, Explain Differential geometry, Define Differential geo...
#Differentialgeometry #audioversity
~~~ Differential geometry ~~~
Title: What is Differential geometry?, Explain Differential geometry, Define Differential geometry
Created on: 2018-11-10
Source Link: https://en.wikipedia.org/wiki/Differential_geometry
------
Description: Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
------
To see your favorite topic here, fill out this request form: https://docs.google.com/forms/d/e/1FAIpQLScU0dLbeWsc01IC0AaO8sgaSgxMFtvBL31c_pjnwEZUiq99Fw/viewform
------
Source: Wikipedia.org articles, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Support: Donations can be made from https://wikimediafoundation.org/wiki/Ways_to_Give to support Wikimedia Foundation and knowledge sharing.
#Differentialgeometry #audioversity
~~~ Differential geometry ~~~
Title: What is Differential geometry?, Explain Differential geometry, Define Differential geometry
Created on: 2018-11-10
Source Link: https://en.wikipedia.org/wiki/Differential_geometry
------
Description: Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
------
To see your favorite topic here, fill out this request form: https://docs.google.com/forms/d/e/1FAIpQLScU0dLbeWsc01IC0AaO8sgaSgxMFtvBL31c_pjnwEZUiq99Fw/viewform
------
Source: Wikipedia.org articles, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Support: Donations can be made from https://wikimediafoundation.org/wiki/Ways_to_Give to support Wikimedia Foundation and knowledge sharing.
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the w...
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
#differentialgeometrylecture
#whatisdifferentialgeometry
#historyofdifferentialgeometry
This video is an extract of my original video 'What Is Differential Geo...
Full episode with Gilbert Strang (Nov 2019): https://www.youtube.com/watch?v=lEZPfmGCEk0
New clips channel (Lex Clips): https://www.youtube.com/lexclips
Once it...
Full episode with Gilbert Strang (Nov 2019): https://www.youtube.com/watch?v=lEZPfmGCEk0
New clips channel (Lex Clips): https://www.youtube.com/lexclips
Once it reaches 20,000 subscribers, I'll start posting the clips there instead.
(more links below)
For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
Clip from full episode: https://www.youtube.com/watch?v=lEZPfmGCEk0 If you enjoy these clips, subscribe to the new clips channel (Lex Clips): https://www.youtube.com/lexclips Once it reaches 20,000 subscribers, I'll start posting the clips there instead. For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
(more links below)
Podcast full episodes playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4
Podcasts clips playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41
Podcast website:
https://lexfridman.com/ai
Podcast on Apple Podcasts (iTunes):
https://apple.co/2lwqZIr
Podcast on Spotify:
https://spoti.fi/2nEwCF8
Podcast RSS:
https://lexfridman.com/category/ai/feed/
Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times.
Subscribe to this YouTube channel or connect on:
- Twitter: https://twitter.com/lexfridman
- LinkedIn: https://www.linkedin.com/in/lexfridman
- Facebook: https://www.facebook.com/lexfridman
- Instagram: https://www.instagram.com/lexfridman
- Medium: https://medium.com/@lexfridman
- Support on Patreon: https://www.patreon.com/lexfridman
Full episode with Gilbert Strang (Nov 2019): https://www.youtube.com/watch?v=lEZPfmGCEk0
New clips channel (Lex Clips): https://www.youtube.com/lexclips
Once it reaches 20,000 subscribers, I'll start posting the clips there instead.
(more links below)
For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
Clip from full episode: https://www.youtube.com/watch?v=lEZPfmGCEk0 If you enjoy these clips, subscribe to the new clips channel (Lex Clips): https://www.youtube.com/lexclips Once it reaches 20,000 subscribers, I'll start posting the clips there instead. For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
(more links below)
Podcast full episodes playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4
Podcasts clips playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41
Podcast website:
https://lexfridman.com/ai
Podcast on Apple Podcasts (iTunes):
https://apple.co/2lwqZIr
Podcast on Spotify:
https://spoti.fi/2nEwCF8
Podcast RSS:
https://lexfridman.com/category/ai/feed/
Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times.
Subscribe to this YouTube channel or connect on:
- Twitter: https://twitter.com/lexfridman
- LinkedIn: https://www.linkedin.com/in/lexfridman
- Facebook: https://www.facebook.com/lexfridman
- Instagram: https://www.instagram.com/lexfridman
- Medium: https://medium.com/@lexfridman
- Support on Patreon: https://www.patreon.com/lexfridman
#howtolearndifferentialgeometry
#differentialgeometrylecture
#differentialgeometrymscmaths
How to learn differential geometry? In this differential geometry le...
#differentialgeometrymscmathematics
#differentialgeometry
#howtolearndifferentialgeometry
How to learn differential geometry? What are the best books for diffe...
Continuing development of differential geometry with vector components and derivation of a formula for the metric tensor in oblique coordinates. The method is easily generalilzed to higher dimensions. Some notes on vector notation and the Einstein summation convention is introduced.
#Differentialgeometry #audioversity
~~~ Differential geometry ~~~
Title: What is Differential geometry?, Explain Differential geometry, Define Differential geometry
Created on: 2018-11-10
Source Link: https://en.wikipedia.org/wiki/Differential_geometry
------
Description: Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
------
To see your favorite topic here, fill out this request form: https://docs.google.com/forms/d/e/1FAIpQLScU0dLbeWsc01IC0AaO8sgaSgxMFtvBL31c_pjnwEZUiq99Fw/viewform
------
Source: Wikipedia.org articles, adapted under https://creativecommons.org/licenses/by-sa/3.0/ license.
Support: Donations can be made from https://wikimediafoundation.org/wiki/Ways_to_Give to support Wikimedia Foundation and knowledge sharing.
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
Full episode with Gilbert Strang (Nov 2019): https://www.youtube.com/watch?v=lEZPfmGCEk0
New clips channel (Lex Clips): https://www.youtube.com/lexclips
Once it reaches 20,000 subscribers, I'll start posting the clips there instead.
(more links below)
For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
Clip from full episode: https://www.youtube.com/watch?v=lEZPfmGCEk0 If you enjoy these clips, subscribe to the new clips channel (Lex Clips): https://www.youtube.com/lexclips Once it reaches 20,000 subscribers, I'll start posting the clips there instead. For now, new full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days.
(more links below)
Podcast full episodes playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4
Podcasts clips playlist:
https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41
Podcast website:
https://lexfridman.com/ai
Podcast on Apple Podcasts (iTunes):
https://apple.co/2lwqZIr
Podcast on Spotify:
https://spoti.fi/2nEwCF8
Podcast RSS:
https://lexfridman.com/category/ai/feed/
Gilbert Strang is a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world. His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times.
Subscribe to this YouTube channel or connect on:
- Twitter: https://twitter.com/lexfridman
- LinkedIn: https://www.linkedin.com/in/lexfridman
- Facebook: https://www.facebook.com/lexfridman
- Instagram: https://www.instagram.com/lexfridman
- Medium: https://medium.com/@lexfridman
- Support on Patreon: https://www.patreon.com/lexfridman
In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface (such as the sphere or the cylinder) at a point can be rolled around the surface to obtain the tangent plane at other points.
Properties
The tangential contact between the surfaces being rolled over one another provides a relation between points on the two surfaces. If this relation is (perhaps only in a local sense) a bijection between the surfaces, then the two surfaces are said to be developable on each other or developments of each other. Differently put, the correspondence provides an isometry, locally, between the two surfaces.
In particular, if one of the surfaces is a plane, then the other is called a developable surface: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.
Flat connections
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection on the surface (it provides an example of parallel transport along a curve), and a developable surface is one for which this connection is flat.