MAT432 Differential Geometry: Metric Spaces, Curves in Euclidian 3 space (E3): Curvature, torsion, fundamental theorem of ordinary differential equations, fundamental existence theorem for space curves. Surfaces in E3: geometry on a surface, Inverse Function Theorem, Implicit Function Theorem, and Gauss curvature. Coordinate charts, Fubini's Theorem, orientation and an introduction to Riemannian Geometry. 4 credits
MAT732: Differential Geometry Curves in E3, curvature, torsion, fundamental existence theorem for space curves, geometry of a surface, inverse and implicit function theorems, Gauss curvature, and Minimal Surfaces. 4 credits
Prerequisite: Linear Algebra MAT313 and Vector Calculus MAT226 and Analysis MAT320 is recommended
Course Webpage: https://sites.google.com/site/professorsormani/teaching/Diff-Geom-2020
Professor Sormani: google "Sormani Math Lehman” for my page
Contact: sormanic@gmail.com (do not call the office and leave messages)
Class Meets: M/W 6:00-7:40 pm Gillet 305
Office Hours: M/W 7:40-9:10 pm in the classroom
Grading Policy:
Expectations: Students are expected to learn both the mathematics covered in class and the mathematics in the textbook and other assigned reading. Completing homework is part of the learning experience. Students should review topics from prior courses as needed using old notes and books. Students are not required to learn MATLAB but it is good to have it as a skill on a resume.
Homework: Approximately four hours of homework will be assigned in each lesson as well as additional review assignments over weekends. The reading includes material not covered in class. Note that a single problem may take an hour. In the schedule below, homework is written below the lesson when it is assigned and starred problems must be submitted. Homework is worth 40% of the grade. Each problem with a single star is worth 1 point towards the 40 points and each with two stars is worth 2 points towards the 40 points. Additional *ed problems that are completed will count as extra credit. Partial credit will be awarded for imperfect work. Resubmission of homework is allowed after feedback to improve your grade on that problem (the original graded work must be included with the resubmission). Late homework will count as resubmitted work that originally had a zero. Your grade on each problem will be the average of the score on the original submission and the resubmission. Note that there are forty starred problems before the midterm exam, and that the problems later in the course will be more difficult, so try to get as much credit as possible for the homework before the midterm exam. You may consult sources for help with homework but you must give credit to those sources by name if it is a person and by full reference if it is a book or website. Students may use MATLAB to help with homework graphing and computations but must teach themselves MATLAB.
Exams: There is a midterm exam (20%) and a final exam (40%) If an exam needs to be retaken because it was missed or due to poor performance, the score on the retake will be multipled by .8 resulting in a maximum score of only 80%. This helps a student avoid failing a course but does not help with a earning an A or a B. The exams may include examples from the reading and homework that were not covered in class. There will be many short answer questions on the exams to confirm that each student has a strong understanding of the material. Long proofs and difficult calculations are on the graded homework and will not be expected to be done in the short time allowed for the exams. There are no quizzes in this course.
Textbook: Bloch, A First Course in Geometric Topology and Differential Geometry, ISBN 0-8176-3840-7, Chapters I, IV, V, and VI
MATLAB: may be used for computations and graphing in this course or the work may be done by hand.
CUNY has MATLAB available here and MATLAB has plenty of help
and Calculus Review Courses using MATLAB
Other resources: Exercises in Computational Mathematics with MATLAB
and Modeling of Curves and Surfaces with MATLAB®
Tutoring: There is no tutoring for this course, but you may stop by professor's office hours before and after class.
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disabipty Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.
Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged for you.
Respect: Everyone will treat each other with respect and dignity. Let me know if you have concerns.
Playlists:
Continuity and Limits in Metric Spaces
Derivatives in Higher Dimensions
Diffeomorphisms and Inverse Function Theorem
Tangent Vectors and Covariant Differentiation
Areas of Parametrized Surfaces and Wormholes - Dr. Jorge Basilio
At the end of the course students should be able to:
1. Write Proofs that sets in metric spaces are open or closed
2. Compute the curvature and torsion of a space curve
3. Compute the change of coordinates, the Jacobian, normal, and area of a parametrized surface
4. Apply the Implicit and Inverse Function Theorems
5. Find the First and Second Fundamental Forms
6. Compute Gauss Curvature
7. Compute Mean Curvature and Verify that a Surface is a Minimal Surface
8. Plot surfaces using Maple or Matlab
Schedule:
Jan 27 Lesson 1: An Introduction to Differential Geometry (photos of boards) (Differentiation Videos Part 1 Part 2 Part 3)
A review of the aspects of Vector Calculus applied in Differential Geometry, review Precalculus notions of one to one and onto and inverse and unit circle, Review Calc I notion of Derivative, derivative of sine and cosine, Increasing, Decreasing, Derivative of Inverse, Rephrase Calc I definition of derivative as approximately a tangent line, review implicit differentiation of a circle, parametrized circle, derivative of a circle giving tangent to a circle, functions of two variables, derivative as an approximate tangent plane, gradient of a function, paraboloid, gradient perpendicular to a level set, functions of three variables like a sphere, parametrization of a sphere using different charts, all giving the same notion of a tangent plane...
Sign into CUNY MATLAB and do the beginning (1-3) of the 2 hour MATLAB onramp course, and try out Easy Plotting of Functions in 2D and 3D adjusting the examples.
Jan 29 Lesson 2: An Introduction to Balls in Metric Spaces and Euclidean Spaces (photos of boards) (videos MetricSpacesPart1 and MetricSpacesPart2)
Prove the following and plot the sets by hand or with MATLAB using Plot Circle 2D and Plotting Planes and Spheres
*(1) The ball about (0,3) of radius 2 is a subset of the upper half plane={(x,y): y>0}
*(2)(a) If d(p,q)= D and D+r<R then B(p,r) is a subset of B(q, R) (b) The ball about (3,4) of radius 2 is a subset of the ball about (0,0) of radius 8.
*(3) the ball about (3,4) of radius 5 is a subset of the plane with the origin removed.
*(4) (a) the ball around (2,3,5) of radius 1 is a subset of {(x,y,z): x>0, y>0, z>0}.
*(4)(b) the ball around (2,3,5) of radius 1 is a subset of [0,4]x[0,5]x[0,6] (see the last photo of today for a hint)
Feb 3 Lesson 3: Lines and Curves in Euclidean Space 4.3 (photos of boards)
Read 4.3 fplot (we will do 4.2 later), Do 4.3.1*, 4.3.6*, 4.3.7*, 4.3.8*, 4.3.9
Students who are trying MATLAB be sure to do the beginning (1-3) of the 2 hour MATLAB onramp course, and try out Easy Plotting of Functions in 2D and 3D adjusting the examples. If you are having trouble, Prof Chen-Yun Lin can see you during her Mon/Wed office hours 2:30-3:30pm in Gillet 230a.
Feb 5 Lesson 4: Open sets in Euclidean Space and in Metric Spaces 1.2 (photos of boards)
Video of Open Set Review Video of Open Set Proofs Part I Video of Open Set Proofs Part II
Read pages 2-7 in 1.2 and a guide to open set proofs, do 1.2.1 (1)**, 1.2.1 (2)**,
Students who have already learned open sets should do more problems from this section.
Feb 10 Lesson 5: Conic Sections (video) (photos of boards)
This homework is due on Wed Feb 19 (note that HW from Lessons 3 and 4 is graded and on my office door 200A Gillet Hall). in addition to doing this homework review what you have learned about planes and lines from vector calculus and linear algebra. In the assignment below x^2 is x squared.
*(1)(a) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=c}. (b) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): x=a}. (c) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): y=b}.
*(2) Find a formula for the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=kx+b} assuming both b and k are positive. (a) For what value of k is this an hyperbola and for what value of k is this an ellipse? (b) Plot one of each.
*(3) Look up the geometric proof involving a sphere that a parabola is a conic section and cite ref.
*(4) (a) Prove that if p=(x,y) is in the set where y<x and if r=distance from p to the line y=x then the ball about p of radius r does not intersect with the line y=x.
*(4)(b) Prove that the set where y<c is an open set.
Lehman Closed Feb 12 and Feb 17
Feb 19 Lesson 6: Unions and Intersections of sets 1.2 (photos of boards and sample exam questions)
Read pages 7-11 of 1.2, read the proof of Lemma 1.2.3 closely
Do *1.2.2, *1.2.3, *1.2.4, *1.2.8
Feb 24 Lesson 7: Smooth curves 4.4 Tangent, Normal, Binormal (photos of boards and sample exam questions)
Read 4.4 (we will do 4.2 later), Do *(1) 4.4.1, *(2) 4.4.2, *(3) 4.4.4
*(4) (a) Find the formula for the distance from a point p to the line y=mx (b) prove that the set U={(x,y): y<mx} is an open set
*(5) Prove that the line L={(0,0,z): z in Reals}which is the z axis is not an open set.
Feb 26 Lesson 8: Limits and Continuous Maps 1.3 (photos of boards and HW review)
Videos of Continuity Part 1 Continuity Part 2 Contunuity Part 3 Limits Also ContinuityPart4 and ContinuityPart5
Read 13-17, focus on Example 1.3.4,
*(1) use an epsilon delta proof to show f(x)=3x+7 is continuous at x=2 (be sure to give a precise formula for delta),
*(2) g(x)=cos(x) is continuous at any x0
March 2 Lesson 9: Curvature and Torsion and Frenet-Serret Formulas 4.5 (video) (photos of boards)
Read Wikipedia and Read 167-168 then Read 4.5, Do * 4.5.3, *4.5.4,
We will do the rest of 4.2 later to rigorously prove the Fundamental Theorem of Curves
The Frenet Frame was applied by NASA Mathematician Katherine Johnson to compute the path of the spacecraft on its way to the Moon. See the movie Hidden Figures for a nice explanation of this as well as the story of her life at NASA.
The Frenet Frame has also been applied to study proteins (see for example this article).
March 4 Lesson 10: Open sets, Closed sets, and Continuity (photos of boards)
Video of Closed Sets Video of products of intervals
In class Extra Credit Quiz: * Open set Proof * Find TN B
HW:* (1) (a) Prove (2,4) is open (b) Prove [2,4) is not open (c) Prove [2,4] is closed
*(2) (a) Prove f(x,y) = x is continuous *(2)(b) Prove f(x,y,z)=(x,y) is continuous
*(3) Prove f(t)=(x(t),y(t)) is continuous iff x(t) and y(t) are continuous
*(4) Prove that if f:X to Y is continuous and U subset Y is open
then the preimage of U={x: f(x) in U} is an open set in X.
!!! Until Further Notice all Lessons are Online not at Lehman !!!
March 9 Lesson 11: MATLAB
Lesson: Sign into CUNY MATLAB and complete the beginning (1)-(3) of the 2 hour MATLAB onramp course, and try out Easy Plotting of Functions in 2D and 3D adjusting the examples for fplot and plotfunct3D to plot the following:
HW *(1a) Use fplot command to plot f(t)=(t,0,0) for t in [0,1] *(1b) Plot f(t)=(1-t,2t,3) for t in [0,1] Note these are line segments!
*(2) Use the plotfunc3D command to plot the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): z=c} for c=0 and c=1. Note you need to solve for z in the formula for a cone to plot it using this command.
*(3) Plot the intersection of a cone {(x,y,z): x^2+y^2=z^2} with a plane {(x,y,z): x=a} for a=0 and a=1.
March 11 Lesson 12: Existence Theorem for Space Curves (lesson photos and video link)
HW *(1a) Find T N B for (3s/5,4sin(s)/5,4cos(s)/5) *(1b) Find T N B for (3t^2/5,4sin(t^2)/5, 4cos(t^2)/5) *(1c) How are these related?
*(2) Suppose you have a curve c(t) and know T N B for c(t). Now consider t as a function of s t=t(s) and find T N B of c(t(s)) at s=s_0. How is this related to TNB of the original curve c at time t=t(s_0)?
*(3) Suppose you have a curve c(t) and know T N B for c(t). Now consider a second curve C(t)=Mc(t) where M is a 3x3 matrix with constant entries. Find T N B for the curve C using formulas depending on T N and B of the original curve c and the matrix M.
*(4) Do the EC inside the lesson photos
March 16 Lesson 13: Review for Midterm Exam: Solutions to the Practice Exam
Sample Midterm with Solutions (but try problems before looking at the solutions)
Video of Open Set Review Video of Open Set Proofs Part I Video of Open Set Proofs Part II Video of Closed Sets
Video of Continuity Part 1 Continuity Part 2 Contunuity Part 3 and Limits
Don't forget to also review finding T N and B Review of Frenet-Serrat
This exam is on Lessons 1-12. You should be able to compute the tangent vector, normal vector, binormal vector, torsion, and curvature of a curve in Euclidean space. You should also be able to identify sets in a plane which are open, not open, and closed without providing proofs. You must also know the definitions of unions and intersections. You will need to know proving techniques including quantifiers, first lines of proofs by contradiction, how to prove one set is a subset of another, how to prove something is an open set (drawing balls about three points to see openness or choosing a bad point), how to prove one set is inside another, how to prove a map is continuous using epsilon deltas. There will not be full length proofs on the exam, just questions regarding first and last steps and such things, what are the definitions, how to draw a diagram that will help you select a ball and so on. Sample Exam Problems will be distributed in class.
March 18 Lesson 14: Midterm Exam is two hours and will be emailed to you when you are ready any time after 5 pm and before 7pm. You email me that you are ready for your individual exam vouching that you have watched the review lesson videos: Video of Open Set Review Video of Open Set Proofs Part I Video of Open Set Proofs Part II Video of Closed Sets You may consult what you wish but do not waste time looking things up. This is a speed test of short answers.
March 23 Lesson 15: Defining the Derivative of a Function, Diffeomorphisms and Chain Rule
Link to Lesson 15 videos, classwork, and homework (watch and do classwork on 3/23-3/24, hw due Mon 4/6 at 6pm.)
March 25 Lesson 16: Diffeomorphisms, Spherical Coordinates and Linear Transformations
Link to Lesson 16 videos, classwork, and homework (watch and do classwork on 3/25-26 hw due Mon 4/6 at 6pm.)
!!! Recalibration Period for Educational Equity March 27-April 2 !!!
April 6 Lesson 17: Inverse Function Theorem 4.2
Link to Lesson 17 videos, classwork, and homework (watch and do classwork on 4/6-7 hw due Mon 4/13 at 6pm.)
Tuesday April 7 Lesson 18: Implicit Function Theorem 4.2
Link to Lesson 18 videos and classwork (watch and do classwork on 4/7-4/8 hw due Mon 4/20 at 6pm.)
Spring Break April 8-10
April 13 Lesson 19: Application and Proof of the Implicit Function Theorem
Link to Lesson 19 videos and classwork (watch and do classwork on 4/13-4/14 hw due Mon 4/20 at 6pm.)
April 15 Lesson 20:
Students may take Parts I and II of the Final either at 6-7 pm or 9-10 pm (based on Lessons 15-17)
Part I Find Df and verify the chain rule for a function as in Derivative Video Part 7
Part II Investigate the Inverse Function Theorem as in Inverse Function Theorem Part 2
April 20 Lesson 21: Smooth Surfaces and Submanifolds 5.1-5.3
Link to Lesson 21 photos, videos explaining photos, classwork and HW
(watch and do classwork April 16-25, HW due Mon April 27)
April 22 Lesson 22: Tangent and Normal Vectors 5.3-5.4
Link to Lesson 22 photos, videos explaining photos, classwork and HW
April 27 Lesson 23: First Fundamental Form 5.5
Link to Lesson 23 photos, videos explaining photos, classwork and HW
April 29 Lesson 24: Differentiation, Lagrange Multipliers, and Vector Fields 5.5-5.6
Link to Lesson 24 photos, videos explaining photos, classwork and HW
May 4 Lesson 25: Covariant Derivatives and Christoffel Symbols 5.5-5.6
Link to Lesson 25 photos, videos explaining photos, classwork and HW
May 6 Lesson 26: Length, Area and Fubini's Theorem 5.8
Guest Speaker: Dr. Jorge Basilio. Areas of Parametrized Surfaces and Wormholes
Lesson Notes with videos by Jorge Basilio
Students interested in possibly pursuing a doctorate in mathematics should contact Prof. Sormani.
HW: Find the area of the image of your patch in your submanifold. To do this choose a domain which is bounded.
May 11 Lesson 27: Second Fundamental Form, Orientation, and Gauss Curvature 6.2
Link to Lesson 27 photos, videos explaining photos, classwork and HW
May 13 Lesson 28: Mean Curvature and Minimal Surfaces 6.5
Link to Lesson 28 photos, videos explaining photos, classwork and HW
The Final is Friday May 22 6-8 pm or 9-11pm or Tuesday May 26 6-8 pm or 9-11 pm. Or different parts at different times. Let me know which time is best for you. Here's a friendly farewell video. Students who need incompletes should still try to complete PartsI-III and possibly even Part IV on the day of the final.
Review for the Final (including links to videos):
I Find Df and verify the chain rule for a function as in Derivative Video Part 7
II Investigate the Inverse Function Theorem as in Inverse Function Theorem Part 2
III Apply the Implicit Function Theorem to verify that a given set is a surface or submanifold as in SurfacePart5
IV Find the tangent vectors, first fundamental form, and area of a surface as in FFFPart2, FFFPart6b, and BasilioExamples
V Compute the mean and Gauss curvature of a surface as in CurvaturePart8, CurvaturePart10b, and CurvaturePart12b
It is important to practice taking partial derivatives from vector calc and finding eigenvalues from linear algebra.
Every student will have a personalized exam and will be sent each part to complete in 30 minutes separately. Students may consult notes, the internet, books, and use software. However students may not consult with another person in the class or online for assistance with the exam.
June schedule for students who need to complete the course or wish to do Research in July:
June 1 Monday at midnight:
submit Lessons 22, 23, and 26.
June 8 Monday at midnight:
submit Lessons 24, 25, 27, and 28
June 15 Monday at midnight:
Last change to resubmit Lessons 22 - 28 to take
Extra Lessons 29-30 (see below) for Extra Credit.
June 22 Monday 6-8 pm and 9-11 pm Final
students who are not ready will privately arrange a date with me before the end of December 2020
or they can consider the credit/no credit option.
June 25 is the deadline for switching a grade to Credit/No Credit (link here)
June 29 Monday at midnight
submit extra Lessons 29-32 if you wish to do the July research.
Extra Lessons after the final to prepare for July Research and Doctoral Study:
Lesson 29: Hessians and Gradients and the Second Derivative Test
Link to Lesson 29 photos, videos explaining photos, classwork and HW
Lesson 30: The Laplacian, Eigenfunctions, and the Laplace Spectrum
Link to Lesson 30 photos, videos, classwork, and HW
Lesson 30A: Rayleigh Quotient and Cheeger's Constant
Lesson 31: Diffusion Maps
Link to Lesson 31 photos, videos, classwork, and HW
Lesson 32: MATLAB and data points on Manifolds
Link to Lesson 32 photos, videos and HW
Lesson 33: Graph Laplacian and Diffusion Map (be sure to complete Lesson 31 first)
Link to Lesson 33 photos, videos, and HW
Extra Lessons on Riemannian Geometry for those who wish to go further:
Lesson A1: Manifolds and Transition Maps
Lesson A2: Riemannian Manifolds
Lesson A4: Submanifolds of Riemannnian Manifolds
Extra Lessons on other subjects for those who wish to go further:
Lesson 34: Gauss Bonnet Theorem (link)
Lesson 35: Completeness and Compactness (for students that have taken Analysis)
Lesson 36: Geodesics 7.1-7.3
Lesson 37: Minimal Surfaces, Calculus of Variations, and Scalar Curvature
Lesson 38: Contraction Mapping Theorem and Proof of Inverse and Implicit Function Theorems
Lesson 39: Applications to Biology