- published: 10 Jun 2018
- views: 160446
-
remove the playlistDifferential Geometry
- remove the playlistDifferential Geometry
Synthetic differential geometry
In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used.
Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry. For example, the meaning of what it means to be natural (or invariant) has a particularly simple expression, even though the formulation in classical differential geometry may be quite difficult.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Synthetic_differential_geometry
Differential geometry
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 characteristic of this field.
History of Development
Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in Calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Differential_geometry
Podcasts:
-
Introduction to Differential Geometry: Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves. After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector. Questions/requests? Let me know in the comments! Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost. Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ Patreon: https://www.patreon.com/user?u=4354534 Twitter: https://twitter.com/FacultyOfKhan Specia...
published: 10 Jun 2018 -
Differential Geometry - Claudio Arezzo - Lecture 01
published: 17 Nov 2016 -
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 -
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 06 Feb 2021 -
Lecture 1: Overview (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 04 Feb 2021 -
Curvature: Intuition and Derivation | Differential Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t. I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal. Questions/feedback? Let me know in the comments! Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x Patreon: https://www.patreon.com/user?u=4354534 Twitter: https://twitter.com/FacultyOfKhan Special Thanks to my Patrons: Cesar Garz...
published: 25 Mar 2020 -
Lecture 5: Differential Forms (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
published: 12 Feb 2021
developed with YouTube
10:25
Introduction to Differential Geometry: Curves
- Order: Reorder
- Duration: 10:25
- Uploaded Date: 10 Jun 2018
- views: 160446
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, whi...
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
https://wn.com/Introduction_To_Differential_Geometry_Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
1:29:40
Differential Geometry - Claudio Arezzo - Lecture 01
- Order: Reorder
- Duration: 1:29:40
- Uploaded Date: 17 Nov 2016
- views: 159616
- published: 17 Nov 2016
- views: 159616
51:32
Differential Geometry | Math History | NJ Wildberger
- Order: Reorder
- Duration: 51:32
- Uploaded Date: 07 May 2012
- views: 167147
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
https://wn.com/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 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
- published: 07 May 2012
- views: 167147
47:44
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
- Order: Reorder
- Duration: 47:44
- Uploaded Date: 06 Feb 2021
- views: 46689
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_2B_Introduction_To_Manifolds_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 06 Feb 2021
- views: 46689
1:07:46
Lecture 1: Overview (Discrete Differential Geometry)
- Order: Reorder
- Duration: 1:07:46
- Uploaded Date: 04 Feb 2021
- views: 53198
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_1_Overview_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 04 Feb 2021
- views: 53198
8:34
Curvature: Intuition and Derivation | Differential Geometry
- Order: Reorder
- Duration: 8:34
- Uploaded Date: 25 Mar 2020
- views: 20192
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve para...
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
https://wn.com/Curvature_Intuition_And_Derivation_|_Differential_Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
- published: 25 Mar 2020
- views: 20192
45:20
Lecture 5: Differential Forms (Discrete Differential Geometry)
- Order: Reorder
- Duration: 45:20
- Uploaded Date: 12 Feb 2021
- views: 28232
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
https://wn.com/Lecture_5_Differential_Forms_(Discrete_Differential_Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
- published: 12 Feb 2021
- views: 28232
10:25
Introduction to Differential Geometry: Curves
In this video, I introduce Differential Geometry by talking about curves. Curves and surfa...
published: 10 Jun 2018
Introduction to Differential Geometry: Curves
Introduction to Differential Geometry: Curves
- Report rights infringement
- published: 10 Jun 2018
- views: 160446
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is why I'm introducing this series by defining curves.
After defining level curves, parametrized curves, and tangent vectors, I solve a short example where I convert a level curve to a parametrized curve and then find its tangent vector.
Questions/requests? Let me know in the comments!
Pre-requisites: A background in Multivariable Calculus (Calculus 3) is helpful, but even if you know the material until Calculus 2, you probably still won't be lost.
Lecture Notes: https://drive.google.com/open?id=1CirfXRYfjS8eKB7TVwEWkAT-8nTzpFEQ
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Jose Lockhart
- Yuan Gao
- James Mark Wilson
- Marcin Maciejewski
- Sabre
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
1:29:40
Differential Geometry - Claudio Arezzo - Lecture 01
published: 17 Nov 2016
Differential Geometry - Claudio Arezzo - Lecture 01
Differential Geometry - Claudio Arezzo - Lecture 01
- Report rights infringement
- published: 17 Nov 2016
- views: 159616
51:32
Differential Geometry | Math History | NJ Wildberger
Differential geometry arises from applying calculus and analytic geometry to curves and su...
published: 07 May 2012
Differential Geometry | Math History | NJ Wildberger
Differential Geometry | Math History | NJ Wildberger
- Report rights infringement
- published: 07 May 2012
- views: 167147
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
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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
47:44
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 06 Feb 2021
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
Lecture 2B: Introduction to Manifolds (Discrete Differential Geometry)
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- published: 06 Feb 2021
- views: 46689
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
1:07:46
Lecture 1: Overview (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 04 Feb 2021
Lecture 1: Overview (Discrete Differential Geometry)
Lecture 1: Overview (Discrete Differential Geometry)
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- published: 04 Feb 2021
- views: 53198
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
8:34
Curvature: Intuition and Derivation | Differential Geometry
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed cu...
published: 25 Mar 2020
Curvature: Intuition and Derivation | Differential Geometry
Curvature: Intuition and Derivation | Differential Geometry
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- published: 25 Mar 2020
- views: 20192
In my 5th video on #DifferentialGeometry, I define the #Curvature for both a unit speed curve reparametrized with respect to arc length and a regular curve parameterized by t.
I describe the intuition behind the curvature as the extent to which a curve deviates from a straight line (zero curvature). I then derive the expression for curvature for both a unit speed curve and a regular curve, using the #UnitNormal.
Questions/feedback? Let me know in the comments!
Pre-reqs: the previous videos in my playlist https://www.youtube.com/playlist?list=PLdgVBOaXkb9DJjk8V0-RkXTnD4ZXUOFsc
Lecture Notes: https://drive.google.com/open?id=1_40zI8E2r81zOmb_nkhWbN7q2gSqUc7x
Patreon: https://www.patreon.com/user?u=4354534
Twitter: https://twitter.com/FacultyOfKhan
Special Thanks to my Patrons:
Cesar Garza
Daigo Saito
Alvin Barnabas
Damjan
Yenyo Pal
Lisa Bouchard
Patapom
Gabriel Sommer
Eugene Bulkin
Yiu Chong
René Gastelumendi
45:20
Lecture 5: Differential Forms (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
Fo...
published: 12 Feb 2021
Lecture 5: Differential Forms (Discrete Differential Geometry)
Lecture 5: Differential Forms (Discrete Differential Geometry)
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- published: 12 Feb 2021
- views: 28232
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS
For more information see http://geometry.cs.cmu.edu/ddg
Synthetic differential geometry
In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used.
Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry. For example, the meaning of what it means to be natural (or invariant) has a particularly simple expression, even though the formulation in classical differential geometry may be quite difficult.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Synthetic_differential_geometry
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