Differential geometry with finite fields | Differential Geometry 7 | NJ Wildberger
25.4 هزار بار بازدید -
11 سال پیش
-
With an algebraic approach to
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
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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/...
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/... 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 Patreon: njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
MathHistory: A course in the History ...
WildTrig: Intro to Rational Trigonometry
Math Foundations
Wild Linear Algebra
Famous Math Problems
Box Arithmetic
Elementary Mathematics (K-6) Explained
Year9Maths
Ancient Mathematics
Wild West Banking
Sociology and Pure Mathematics
Sociology and Pure Physics
Old Babylonian Mathematics
Probability and Statistics: an introd...
Boole's Logic and Circuit Analysis
Universal Hyperbolic Geometry
Differential Geometry
Algebraic Topology
MathSeminars
Playing Go
Diffusion Symmetry: A bridge between ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One
Classical to Quantum
Algebraic Calculus Two
Advice for mathematics exploration
Solving Polynomial Equations | Explor...
The Hexagrammum Mysticum (Members only)
Algebraic Calculus and dCB curves | E...
Dynamics on Graphs
Playlist
Triangle Geometry
Explorations with q-series
Six: An elementary course in Pure Mat...
************************
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
************************
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/...
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/... 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 Patreon: njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
MathHistory: A course in the History ...
WildTrig: Intro to Rational Trigonometry
Math Foundations
Wild Linear Algebra
Famous Math Problems
Box Arithmetic
Elementary Mathematics (K-6) Explained
Year9Maths
Ancient Mathematics
Wild West Banking
Sociology and Pure Mathematics
Sociology and Pure Physics
Old Babylonian Mathematics
Probability and Statistics: an introd...
Boole's Logic and Circuit Analysis
Universal Hyperbolic Geometry
Differential Geometry
Algebraic Topology
MathSeminars
Playing Go
Diffusion Symmetry: A bridge between ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One
Classical to Quantum
Algebraic Calculus Two
Advice for mathematics exploration
Solving Polynomial Equations | Explor...
The Hexagrammum Mysticum (Members only)
Algebraic Calculus and dCB curves | E...
Dynamics on Graphs
Playlist
Triangle Geometry
Explorations with q-series
Six: An elementary course in Pure Mat...
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11 سال پیش
در تاریخ 1392/06/07 منتشر شده
است.
25,489
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