Skip to content

Commit

Permalink
Signed-off-by: Silvio Fanzon <silvio.fanzon@gmail.com>
Browse files Browse the repository at this point in the history
  • Loading branch information
@sfanzon
sfanzon committed Sep 26, 2024
1 parent a827594 commit dc113ae
Show file tree
Hide file tree
Showing 2 changed files with 55 additions and 54 deletions.
21 changes: 11 additions & 10 deletions _posts/2024-09-15-Differential-Geometry.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -147,18 +147,19 @@ The module Lecture Notes are available **[here](https://www.silviofanzon.com/202


|**Lesson #**|**Date** | **Time** | **Topics** |
|: ------- |: ------- |:--------- |:--------- |
| 1 | 26/09/24 | 16:00 - 18:00 | Intro: Access Canvas page and Lecture Notes. Briefing on Assignments, Assessment. Curves as Level sets. Parametrized curves. Examples. |
| 2 | 27/09/24 | 09:00 - 10:00 | Plotting curves in Python using Matplotlib and Plotly. Parametrized curves examples. Tangent vector. Definition of Length of $\gamma$ as limit of length of polygonals. |
| 3 | 27/09/24 | 16:00 - 18:00 | Proven that $L(\gamma)$ can be compute by integrating $\| \dot \gamma \|$ when $\gamma$ regular. Computed length of circle and portion of Helix. |
| 4 | 03/10/24 | 16:00 - 18:00 | Arc-Length. Examples. Scalar product in $\mathbb{R}^2$: geometric definition. SP in coordinates and in $\mathbb{R}^n$. Bilinearity, Symmetry, Differentiation of SP. |
|: ------- |: ------- |:--------- |:--------- |
| 1 | 26/09/24 | 16:00 - 18:00 | Intro: Canvas and Lecture Notes. Briefing on Assessment, Timetable, etc. Curves as Level sets. Parametrized curves. Smooth curves. Tangent vector. Examples. Length of $\gamma$ as limit of length of polygonals.
Proof: $L(\gamma) = \int \| \dot \gamma \|$ when $\gamma$ regular. |
| 2 | 27/09/24 | 09:00 - 10:00 | Length of circle and portion of Helix. Arc-Length. Examples. Scalar product in $\mathbb{R}^2$: geometric definition. SP in coordinates and in $\mathbb{R}^n$. Bilinearity, Symmetry, Differentiation of SP. |
| 3 | 27/09/24 | 16:00 - 18:00 | |
| 4 | 03/10/24 | 16:00 - 18:00 | |
| 5 | 04/10/24 | 09:00 - 10:00 | Speed. Reparametrizations. Regular and singular points. Unit speed reparametrization. Thm: Regularity is equivalent to existence of unit speed reparametrization. |
| 6 | 04/10/24 | 16:00 - 18:00 | Theorem: Arc-length as unit speed reparametrization. Periodic curves. Closed curves. Period of a closed curve. Theorem: Period of closed curve exists. Examples. |
| 7 | 10/10/24 | 15:00 - 16:00 | Curvature: motivation with Taylor formula. Curvature for unit speed curves. Curvature of circle of radius $R$ computed via reparametrization to unit speed curve. |
| 8 | 10/10/24 | 16:00 - 18:00 | Vector Product in $\mathbb{R}^3$: Algebraic definition and geometric properties. General formula for curvature of regular curves (no proof for now). Examples. |
| 9 | 11/10/24 | 16:00 - 18:00 | Plane curves. Signed curvature (SC). Geometric meaning of SC. SC characterizes plane curves (no proof). Correction of Homework 1. |
| 10 | 17/10/24 | 16:00 - 18:00 | Hyperbolic functions and their properties. Example of the catenary curve. |
| 11 | 18/10/24 | 09:00 - 10:00 | Torsion for unit speed curves. Torsion, general formula. Example of calculation of curvature and torsion. Frenet Frame. Frenet-Serret equations. |
| 10 | 17/10/24 | 16:00 - 18:00 | Hyperbolic functions and their properties. Example of the catenary curve. |
| 11 | 18/10/24 | 09:00 - 10:00 | Torsion for unit speed curves. Torsion, general formula. Example of calculation of curvature and torsion. Frenet Frame. Frenet-Serret equations. |
| 12 | 18/10/24 | 16:00 - 18:00 | Frenet Frame of Helix. Characterization Theorem of 3D curves. New notations. Proof of general curvature formula. Geometric consequences of Frenet-Serret. |
| 13 | 24/10/24 | 15:00 - 16:00 | Definition of Topology. Trivial, discrete and euclidean topologies. Balls are open in $\mathbb{R}^n$. Closed sets. Topology through closed sets. Zariski topology. |
| 14 | 24/10/24 | 16:00 - 18:00 | Comparing topologies. Cofinite topology. Convergence. Metric spaces. Interior, closure, boundary in general topological spaces. |
Expand All @@ -181,9 +182,9 @@ The module Lecture Notes are available **[here](https://www.silviofanzon.com/202
| 31 | 05/12/24 | 16:00 - 18:00 | Matrix of Weingarten map. Gaussian and mean curvatures $G$ and $H$. Formulas for $G$ and $H$. Principal curvatures and directions. Relationship to $G$ and $H$. Examples.|
| 32 | 06/12/24 | 09:00 - 10:00 | Normal and Geodesic curvatures. Euler's Theorem. Local shape of surface: Elliptic, Hyperbolic, Parabolic and Planar points. Local Structure Theorem. |
| 33 | 06/12/24 | 16:00 - 17:00 | Umbilical points and structure theorem at umbilics. |
| 34 | 12/12/24 | 16:00 - 18:00 | Revision and Exam Preparation. |
| 35 | 13/12/24 | 09:00 - 10:00 | Revision and Exam Preparation. |
| 36 | 13/12/24 | 16:00 - 17:00 | Revision and Exam Preparation. |
| 34 | 12/12/24 | 16:00 - 18:00 | Revision and Exam Preparation. |
| 35 | 13/12/24 | 09:00 - 10:00 | Revision and Exam Preparation. |
| 36 | 13/12/24 | 16:00 - 17:00 | Revision and Exam Preparation. |



Expand Down
Loading

0 comments on commit dc113ae

Please sign in to comment.