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.
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
Infinities and Skepticism in Mathematics: Steve Patterson interviews N J Wildberger
In this special video, Steve Patterson interviews N J Wildberger on a range of foundational issues exploring infinities and the role of skepticism in modern mathematics. Steve is a philosopher who runs a popular podcast called Patterson in Pursuit, and you can find more of his work at
http://steve-patterson.com/podcast/
We cover quite a lot of territory, including the core topic at the heart of the modern foundational difficulties with mathematics, the difference between algorithms and infinite choice, conundrums like the Banach Tarski paradox, Wittgenstein's insights, the role of axiomatics in modern mathematics, the nature of space, the direction of future mathematics, and more!
A big thanks to Daniel Mansfield for setting up the studio and videoing our talk, and to Steve Patterson fo...
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
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
In this special video, Steve Patterson interviews N J Wildberger on a range of foundational issues exploring infinities and the role of skepticism in modern mat...
In this special video, Steve Patterson interviews N J Wildberger on a range of foundational issues exploring infinities and the role of skepticism in modern mathematics. Steve is a philosopher who runs a popular podcast called Patterson in Pursuit, and you can find more of his work at
http://steve-patterson.com/podcast/
We cover quite a lot of territory, including the core topic at the heart of the modern foundational difficulties with mathematics, the difference between algorithms and infinite choice, conundrums like the Banach Tarski paradox, Wittgenstein's insights, the role of axiomatics in modern mathematics, the nature of space, the direction of future mathematics, and more!
A big thanks to Daniel Mansfield for setting up the studio and videoing our talk, and to Steve Patterson for lots of interesting questions and comments. He is coming from philosophy, and I from mathematics, but there is a lot of common ground and understanding. Hope you enjoy the discussion: I certainly did!
************************
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.
In this special video, Steve Patterson interviews N J Wildberger on a range of foundational issues exploring infinities and the role of skepticism in modern mathematics. Steve is a philosopher who runs a popular podcast called Patterson in Pursuit, and you can find more of his work at
http://steve-patterson.com/podcast/
We cover quite a lot of territory, including the core topic at the heart of the modern foundational difficulties with mathematics, the difference between algorithms and infinite choice, conundrums like the Banach Tarski paradox, Wittgenstein's insights, the role of axiomatics in modern mathematics, the nature of space, the direction of future mathematics, and more!
A big thanks to Daniel Mansfield for setting up the studio and videoing our talk, and to Steve Patterson for lots of interesting questions and comments. He is coming from philosophy, and I from mathematics, but there is a lot of common ground and understanding. Hope you enjoy the discussion: I certainly did!
************************
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.
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
In this special video, Steve Patterson interviews N J Wildberger on a range of foundational issues exploring infinities and the role of skepticism in modern mathematics. Steve is a philosopher who runs a popular podcast called Patterson in Pursuit, and you can find more of his work at
http://steve-patterson.com/podcast/
We cover quite a lot of territory, including the core topic at the heart of the modern foundational difficulties with mathematics, the difference between algorithms and infinite choice, conundrums like the Banach Tarski paradox, Wittgenstein's insights, the role of axiomatics in modern mathematics, the nature of space, the direction of future mathematics, and more!
A big thanks to Daniel Mansfield for setting up the studio and videoing our talk, and to Steve Patterson for lots of interesting questions and comments. He is coming from philosophy, and I from mathematics, but there is a lot of common ground and understanding. Hope you enjoy the discussion: I certainly did!
************************
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.
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.