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Math 435: Vector Analysis and Introduction to Differential Geometry, Spring 2023

MWF 11:00-11:50am in KAP 137

Teaching Staff

Instructor Joj Helfer TA Tina Peng
email jhelfer (at) usc (dot) edu email pengtina (at) usc (dot) edu
Office Hours After class in KAP 464-B Office hours MW 8-9:30am (in the Math Center, KAP 263)

You may be looking for the the problem sets, the lecture plan, or the practice exam.

Announcements +open all -close all

Course announcements made on Blackboard will appear here as well.

Finals graded Friady, May 5

Hi all,

The finals have been graded and the solutions are posted on the course webpage.

If you have any questions about the grading (or anything else) or if you have regrade requests, please submit them by the evening of Sunday, May 7 for them to be considered before the final grades are submitted.

Thanks again for a great semester!


Office Hours Monday Saturday, Apr. 29

Hi everyone,

I will have office hours from 8AM-10AM THEN 11AM-12PM on Monday May 1st in the Math Center as well as on zoom via the math center zoom link. This is in addition to the final review session from 10AM-11AM in KAP113 mentioned in the previous announcement. Please email me with any questions. Thank you!


HW13 solutions posted and info about the final on May 3 Friday, Apr. 28

Hello all,

The solutions for homework 13 have been posted.

Also, here is some information (which was also mentioned in class) about the final, which will be in KAP 137 on Wednesday, May 3 at 11am to 1pm:

  • The final will cover all the material from the course, with an emphasis on the second half of the course.
  • Tina will be doing a review session on Monday, May 1 (see her announcement).
  • A practice final and solutions have been posted on the course webpage.
  • As mentioned on the course webpage, the final will be a closed book, closed notes exam, with no calculators or other electronic aids permitted.
  • Please consult the course schedule for the list of topics covered in the course. Most importantly, you should know the definitions of all the concepts we have discussed. You should also be prepared to solve any of the Homework problems.

Please let me or Tina know if you have any questions or concerns about the final.


Final Review Session Tuesday, Apr. 25

Hello everyone,

We will have a final review session Monday May 1st, 10AM to 11AM in KAP113. I will try to zoom the classroom, barring technical difficulties. Please email me with questions. Thank you all so much!


Midterm Review Session Monday, Feb. 27

Hi everyone,

I will be holding a review session for the midterm this week on Thursday (3/2) from 2-3PM at CPA110. We will be going over the practice midterm. Please email me with any questions. Thank you!


Midterm on Friday, March 3 Friday, Feb. 24

Hi all,

Here is some information about the midterm, which will be in class on Friday, March 3.

  • There will be no homework due on March 1. Homework 7 is due on Wednesday, March 8.
  • The midterm will cover everything up to and including the material from the class on February 22. (That is, everything up to and including surfaces, but not including smooth surfaces.)
  • Tina will be doing a review session on Thursday, March 2. She will send out an announcement about it.
  • A practice midterm will be posted on the course webpage.
  • As mentioned on the course webpage, the midterm will be a closed book, closed notes exam, with no calculators or other electronic aids permitted.
  • Please consult the course schedule for the list of topics that have been covered so far. Most importantly, you should know the definitions of all the concepts we have discussed. You should also be prepared to solve any of the problems from Homework 1-6.

Please let me or Tina know if you have any questions or concerns about the midterm.


Official Office Hours Monday, Jan. 23

Hi everyone,

My (Tina’s) office hours for the rest of the semester are:

8:00-9:30AM Monday

8:00-9:30AM Wednesday

I will physically be in the math center during these times every week. Please email me with any questions or concerns. Thank you!

Office Hours This Week Only (Updated) Monday, Jan. 16

Hi everyone,

My office hours for this week only are:

     Tuesday 11:30-1

     Friday 1-2:30

I will be at the math center. Please email me if you have any questions. Again, these times are only for this week. Thank you!


Office Hours Tomorrow Monday, Jan. 16

Hi everyone,

My office hours through the math center have not been established yet. Since there have been some requests to hold office hours before homework is due this week, I’ll be at the math center tomorrow around noon. I might only be able to stick around for an hour, but come by if you’d like to meet me and ask questions. I’ll let everyone know when my official office hours have been decided.


Welcome! Friday, Jan. 6

Hello everyone and welcome to Math 435! I’m looking forward to meeting you all on Monday.

I encourage you to check out the course website at, though we will be going over the information there together on Monday.

One thing, however, that I would like to draw your attention to, is that classes will be in person and attendance is mandatory.

In particular, if you have any concerns about being able to make it on Monday -- or if there is anything else you would like to ask me about -- please email me at jhelfer (at) usc (dot) edu or our TA Tina Peng at pengtina (at) usc (dot) edu.



Differential geometry is a topic at the intersection of multiple fundamental topics: multivariable calculus, differential equations, linear algebra, topology, real analysis, and of course geometry. It is a long journey that takes you everywhere. It requires patience, refection and naturally hard work. Ultimately, it helps to develop a better understanding of how different, powerful mathematical topics and methods come together to deliver some of the finest results in mathematics.

By the end of the semester, you will be familiar with the fundamentals of differential geometry in R2 and R3 and its applications to the study of curves and surfaces. Main objectives for this course are to cover Gauss’ Theorema Egregium and Gauss-Bonnet Theorem. Before getting there, we will have to cover multiple concepts: curves, surfaces, advanced multivariable calculus, curvature, connection, geodesics, and some topology

For a list of topics to be covered, please see the tentative lecture plan below.


The official course text is:

It is available for free from the USC Library .


The prerequisite is Math 226 (Calc III) or Math 227 (Enhanced Calc III). Quoting from the textbook: “the only pre-requisites are a good working knowledge of Calculus (including partial differentiation), Vectors and Linear Algebra (including matrices and determinants)”. Any concepts that are needed but that were not covered in Math 226/227 will be covered in class.


Class attendance is mandatory. If you have any scheduling conflicts, or you think you might be unable to make it to any given class, please inform the instructor or TA as soon as possible.

Classes will be in person. Certain classes may be given remotely over Zoom, in which case you will be notified in advance, and you will be able to access the Zoom meeting via the Blackboard page.

This page, Blackboard, Gradescope

All course information will be available on this webpage.

We will make limited use of other platforms as follows:

Grading scheme

Homework assignments

Homework assignments will be posted here each week.

Problem sets:

Due date Assignment
Wed, Jan. 18 Homework 1. (Solutions)
Wed, Jan. 25 Homework 2. (Solutions)
Wed, Feb. 1 Homework 3. (Solutions)
Wed, Feb. 8 Homework 4. (Solutions)
Wed, Feb. 15 Homework 5. (Solutions)
Wed, Feb. 22 Homework 6. (Solutions)
Wed, Mar. 8 Homework 7. (Solutions)
Wed, Mar. 22 Homework 8. (Solutions)
Wed, Mar. 29 Homework 9. (Solutions)
Wed, Apr. 5 Homework 10. (Solutions)
Wed, Apr. 12 Homework 11. (Solutions)
Wed, Apr. 19 Homework 12. (Solutions)
Wed, Apr. 26 Homework 13. (Solutions)

Homework is to be submitted on Gradescope (you should see the course there if you are enrolled; if you have any problems, please inform the instructor or TA). When you submit on Gradescope, please don’t forget to match your scanned pages with the problems.

Homework must be submitted by the posted due dates. If you expect to have issues submitting the homework on time, or if you are having difficulties with gradescope, please write to the instructor or TA as soon as possible, and attach a scanned copy of your submission (this is a wise practice for any class).


Both exams are closed book, closed notes exams, with no calculators or other electronic aids permitted. The final exam will cover all topics from the semester, but will have greater emphasis on topics developed after the midterm.

Office hours

The TA’s office hours will be held in the Math Center (KAP 263); the time is 8-9:30am on Mondays and Wednesdays.

The instructor’s office hours are Mondays at 1-2pm, and Wednesdays and Fridays at 12-1pm, and are “by appointment”; i.e., if you want to attend office hours on a given day, send an email beforehand, or just let the instructor know at the end of class.

Students with Disabilities

Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. If required, please make sure that the DSP letter (for approved accomodations) is delivered to me as early in the semester as possible. For more details, see the DSP web site here; in particular contact information is here.

Academic Integrity

The instructor strongly adheres to the University policies regarding principles of academic honesty and academic integrity violations, and will strictly enforce these rules. You are encouraged to review those, for instance in SCampus, the Student Guidebook (see e.g., University Governance, Section 11.00 and Appendix A).


This syllabus is not a contract, and the Instructor reserves the right to make some changes during the semester.

Tentative schedule

Note: this schedule is tentative and will be continuously updated to adapt to the pace of the course. Please check back regualrly for updates.

Week Date Material
Mon, Jan. 9
  • Overview of differential geometry
    • Euclid: ~300BC
    • Descartes: ~1637
    • Newton/Leibniz: ~1670
    • (Euler, Lagrange, Monge, ...) 1700s
    • Gauss: 1827: General Investigations on Curved Surfaces
    • Elimination of infinitesimals (Cauchy-Weierstass ~1825-1861) and modern mathematics (set theory; Cantor-Dedekind-Hilbert-Frege-Russell, ~1900)
    • Riemann: 1854
    • Einstein: 1915
    • The modern era of differential/Riemannian geometry (1900s)
  • Brief overview of topics
    • Parametrized curves/surfaces versus level curves/surfaces
    • Tangent vectors/planes
    • Curvature
    • Intrinsic geometry and Theorema Egregrium
    • Geodesics
    • Global topology and Gauss-Bonnet
    • Many other little things along the way, and many examples
  • Parametrized curves (Definition 1.1.1)
  • Level curves and parametrization of (part of) a level curve (p .2)
  • Parametrizing a parabola (Example 1.1.2)
  • Some already-known concepts
    • The set of real numbers and n-tuples of real numbers
    • Functions between sets
    • Set-builder notation
    • Note: mere familiarity with and intuition about these basic concepts is enough; for the concepts introduced in this class, you must know the precise definitions.
1 Wed, Jan. 11
  • Review of two steps involved in finding a parametrization of a curve
    • Definition of the image of a curve: im(γ)={γ(t) | t ∈ (α,β)}
    • The two steps in proving that two steps are equal (⊂ and ⊃)
  • Parametrizing the circle (Example 1.1.3)
  • Parametrizing the astroid (Example 1.1.4)
  • Definition of smooth functions and smooth parametrized curves (p. 4)
    • Smooth functions are closed under basic operations (addition, composition, etc.)
    • From now we assume all parametrized curves are smooth
  • Tangent vector to a curve (Definition 1.1.5)
    • Definition of tangent line
  • A curve with constant tangent vector is a straight line (Proposition 1.1.6)
    • Proof next class
  • Some already-known concepts
    • Derivatives and higher derivatives
    • Limits
    • Rules for derivatives (sum, product, quotient, chain rule)
    • Trigonometric functions, their derivatives, and trig identities
Fri, Jan. 13
  • Proof of Proposition 1.1.6
  • The limaçon (Example 1.1.7)
    • A curve can have two different tangent vectors at the same point at different times
    • Thus it is an abuse of notation to write “the tangent vector at the point γ(t)”, but we do it anyway
  • Arc-length (Definition 1.2.1)
    • Arc-lengths starting at different points differ points differ by a constant
    • The derivative of the arc length is the speed
  • Unit-speed curves (Definition 1.2.3)
    • For a unit speed curve, the parameter is just the arc-length, up to a constant
  • The product rule for derivatives of dot products (p. 11)
  • The tangent vector of a unit-speed curves is orthogonal to its derivative (Proposition 1.2.4)
  • Some already-known concepts
    • Integrals
    • The fundamental theorem of calculus
    • Other integration rules (substitution, integration by parts, etc.)
    • Riemann sums
    • Norm/length of a vector
    • Dot product
    • The norm squared of a vector is its dot product with itself
Mon, Jan. 16 MLK’s birthday. No class!
2 Wed, Jan. 18
  • Reparametrization of a curve (Definition 1.3.1)
    • If γ1 is a reparametrization of γ2, then γ2 is a reparametrization of γ1.
    • Warning: not every smooth bijection has a smooth inverse
  • Reparametrizing the circle (Example 1.3.2)
    • Warning: given a parametrization γ of a level curve C, not every parametrization of C is a reparametrization of γ.
  • Digression on “change of variables”
  • Regular/singular point and regular curve (Definition 1.3.3)
  • Any reparametrization of a regular curve is regular (Proposition 1.3.4)
  • Lemma: any smooth bijection with smooth inverse has a non-vanishing derivative
    • and converesely, any smooth function with non-vanishing derivative is a bijection onto its image and has a smooth inverse
Fri, Jan. 20
  • The concept of “parameter” (see this supplement)
    • The derivative of a curve or function with respect to a parameter
    • A parameter u is a unit-speed parameter if and only if the reparametrization with respect to u is a unit speed curve
  • The arc-length of a curve is a parameter if and only if the curve is regular (Propositions 1.3.6-1.3.7)
  • Up to sign and a constant, the arc-length is the only unit speed parameter (Corollary 1.3.7)
  • A unit speed parametrization can be difficult or impossible to compute
    • Example 1.3.8: logarithmic spiral
    • Example 1.3.9: the twisted cubic
Mon, Jan. 23
  • T-periodic curves and closed curves (Definition 1.4.1)
    • Every curve is 0-periodic
    • A curve is T-periodic if and only if it is (-T)-periodic, so we might as well always assume T≥0.
  • The period of a closed curve (Definition 1.4.2)
  • The length of a closed curve (p. 21)
  • A unit-speed parametrization is closed, and the period is the length (p.21)
  • Self-intersection (Definition 1.4.4)
  • Example: the limaçon has one self-intersection (Example 1.4.5)
  • Smooth multivariable functions (p. 23)
  • Regular level curves (the defining function is smooth with non-vanishing gradient)
  • For any point p on a regular level curve C, there is a regular parametrization of part of C passing through P (Theorem 1.5.1)
3 Wed, Jan. 25
  • Proof of Theorem 1.5.1 (except for the smoothness and regularity of the parametrized curve)
    • If one introduces the notion of a “connected” curve, one can show that for a connected regular level curve, there is a regular parametrization of the whole curve
  • For a regular parametrized curve, there is a piece of it near any point on it that is part of a regular level curve (Theorem 1.5.2)
  • Curvature of unit speed curves (Definition 2.1.1)
  • A circle of radius R has constant curvature 1/R (p. 31)
Fri, Jan. 27
  • Curvature of regular curves (p. 31)
    • Checked that it's independent of the unit-speed parameter used to define it
  • Review of cross products
  • Formula for curvature in terms of first and second derivative of γ (Proposition 2.1.2)
  • Signed unit normal vector and signed curvature (p. 35)
  • Review of rotation matrices
Mon, Jan. 30
  • Definition (2.2.2) and existence and uniqueness (Proposition 2.2.1) of the turning angle
  • The turning angle is equal to the signed curvature (Proposition 2.2.3)
4 Wed, Feb. 1
  • Signed curvature of catenary (Example 2.2.4)
    • Trick: the tangent tan(φ) of the turning angle is the quotient of the components of the tangent vector
  • Total signed curvature of a closed curve (p. 39)
  • The total signed curvature is a multiple of 2π (Corollary 2.2.5)
  • Isometries and direct isometries of the plane (p. 39)
  • Any prescribed signed curvature function determines a unit-speed curve, which is unique up to direct isometry (Theorem 2.2.6)
Fri, Feb. 3
  • A plane curve with non-zero constant curvature is part of a circle (Example 2.2.7)
    • In this case, signed curvature is plus or minus the curvature, since an integer-valued continuous function on a connected interval must be constant
  • A simple signed curvature function can lead to a complicated curve (Example 2.2.8)
  • If the curvature is not non-vanishing, it does not determine the curve up to isometry
    • The example form p. 43
  • In R3, even if the curvature is non-vanishing, it does not determine the curve up to isometry
    • Example: a helix and a circle both have constant curvature, but are obviously not related by an isometry (p. 46)
  • Definition of principal normal vector, binormal vector, and of torsion (pp. 46-47)
  • The unit tangent, principal normal, and binormal vector form an oriented/right-handed orthonormal basis at every point (p. 46)
    • A basis is orthonormal if and only if the matrix with those vectors as columns or rows is orthogonal
    • An ordered basis is oriented if and only if the associated matrix has determinant 1
  • The product rule for cross products (p. 47)
Mon, Feb. 6
  • Formula for torsion in terms of the derivatives of γ (Proposition 2.3.1)
  • (axb)·c is the determinant of the matrix with columns a,b,c
  • Torsion of a helix (Example 2.3.2)
  • Torsion vanishes if and only if the curve lies in a plane (Proposition 2.3.3)
  • Review of equational form of a plane
  • The Frenet-Serret equations (Theorem 2.3.4)
  • Skew-symmetric matrices (p. 51)
5 Wed, Feb. 8
  • The curvature and torsion determine a curve up to isometry (Theorem 2.3.6)
    • Some nice ideas that we used in the proof, but that aren't essential knowledge for this class:
    • A system of equations of the form X'(t)=AX(t) (with A and X matrices) can be solved using matrix exponentials
    • If X'(t) is skew-symmetric for all t, then X(t) is orthogonal for all t (provided X(t0) is orthogonal for some t0)
Fri, Feb. 10
  • Simple closed curves (Definition 3.1.1)
    • The limaçon is closed but not a simple closed curve (Example 3.1.3)
  • The Jordan Curve Theorem: the complement of a simple closed curve is the disjoint union of a bounded "interior" and an unbounded "exterior" (p. 55)
    • Sketch of proof: first prove it for polygons by counting whether there are an even or odd number of points below a given point. And then approximate a general curve by a polygonal curve.
    • For an ellipse, we can prove the theorem directly (Example 3.1.2: f)
  • Definition of "positively-oriented" based on the notion of "interior" coming from the Jordan Curve Theorem (p. 57)
  • Hopf's Umlaufsatz: the total signed curvature of a simple closed curve is ±2π, with the sign given by whether the curve is positively or negatively oriented. (Theorem 3.1.4)
    • A proof along the lines sketched in class can be found here
  • Reminder on double integrals
    • They are first defined on rectangles using Riemann sums
    • They are extended to general bounded regions by multiplying with a characteristic function
    • Fubini's theorem: they can be computed on rectangles, or on the area bounded by the graphs of two functions, using an interated integral
  • Definition of the area bounded by a curve: the integral of the function 1 over its interior.
  • Statement of the Isoperimetric Inequality (Theorem 3.2.2): A≤ℓ2/4π
Mon, Feb. 13
  • Green's theorem (p. 58)
  • The definition of ∫f(x,y)dx + g(x,y)dy over a curve γ
    • It is defined as ∫f(u(t),v(t))u'(t)dt + g(u(t),v(t))v'(t)dt over the period of γ, where u and v are the components of γ
  • Wirtinger's inequality (Proposition 3.2.3, but we used the version here)
    • This involved some Fourier analysis: every smooth 2π-periodic function f(t) is a sum a0/2 + Σk≥1 akcos(kt) + bksin(kt)
    • Also, the integral of f(t)2 is just the sum a02 + Σk≥1 ak2 + bk2
  • Proof of isoperimetric inequality (Theorem 3.2.2, but we followed the proof here)
6 Wed, Feb. 15
  • Intuitive notion of a surface: a subset of R3 that "looks like a piece of R2 in the vicinity of each point
  • Open subsets of Rn (p. 68)
  • Open and closed balls around points (p. 68)
  • Continuous maps between subsets of Rn (with possibly different "n" for the domain and codomain)
  • Facts about continuous maps:
    • A map is continuous if and only if each of its coordinate functions are continuous
    • Continuous maps to R are closed under addition, subtraction, multiplication, division (if the denominator is non-vanishing), and constant maps are continuous
    • The composite of two continuous maps is continuous
    • Any map on X⊂Rn which is the restriction of a smooth (or even once-differentiable in each variable) map on Rn is continuous
    • Summary, "if it looks continuous, it's continuous" and "almost every map you'll come across is continuous"
  • Fact about open sets: any set defined by strict inequalities between continuous functions is open
  • Homeomorphisms (p. 68)
  • Surfaces (Definition 4.1.1)
    • Some auxiliary notions:
    • Open subset of a set S⊂Rn
    • Surface patches/parametrizations
    • Atlases
  • Every plane in R3 is a surface (Example 4.1.2)
Fri, Feb. 17
  • A cylinder is a surface (Example 4.1.3)
  • A sphere is a surface (Example 4.1.4)
  • Warning: Pressley often write "parametrization" when he just mean "continous surjection onto a surface", rather than the official definition: "homeomorphism from an open subset of R2 to an open subset of the surface".
    • In both the case of the cylinder and sphere, we found our atlas of surface patches by starting with a natural "parametrization" of the whole surface, and then restricting the domain so as to make it a homeomorphism.
Mon, Feb. 20 President's day. No class!
7 Wed, Feb. 22
  • Some topology (see this supplement)
  • The circular cone is not a surface (Example 4.1.5)
    • The reason is that any path from the "top half" of the cone to the "bottom half" must pass through the vertex, whereas in any open disk in the plane, any two points can be joined by a path missing the center; this shows that there cannot be a surface patch containing the center.
    • However, if we remove the vertex, then it is a surface.
    • Also, if we only take "one half" of the cone, it is also a surface (though not a smooth surface, a concept we will come to soon)
  • Transition maps between surface patches (p. 74)
Fri, Feb. 24
  • Smoothness and partial derivatives of multivariable functions (p. 76)
  • Regular/allowable surface patches (Definition 4.2.1)
  • Smooth surfaces (Definition 4.2.2)
  • The plane, cylinder and sphere are smooth (Examples 4.2.3-4.2.5)
  • The transition maps of a smooth surface are smooth (Proposition 4.2.6); we will see the proof later
  • A diffeomorphism between open subsets of Rn is a smooth bijection with smooth inverse
    • Given a diffeomorphism Φ:U→U' and a map σ:U→R^m, the reparametrization of σ with respect to Φ is the map σ∘Φ-1:U'→R^m
  • Any reparametrization of a regular surface patch is regular (Proposition 4.2.7); we will see the proof next class
  • The multivariable chain rule; next class, we will discuss this supplement
Mon, Feb. 27
  • Partial derivatives with respect to an arbitrary coordinate system (see this supplement)
  • Proof of Proposition 4.2.7 (any reparametrization of a regular surface patch is regular)
    • Reminder: the determinant of an invertible matrix is non-zero
    • We also used that (by the matrix version of the chain rule) the derivative of a diffeomorphism at each point is an invertible matrix
    • There is a more direct proof of the proposition, using that if a matrix has linearly independent columns, so does any product of it with an invertible matrix
  • Corollary of Proposition 4.2.6 (p. 79): any two surfaces patches, restricted to their overlapping regions, are reparametrizations of one another
  • Convention from now on (p. 79): unless we say otherwise, we will assume all surfaces are smooth and connected
8 Wed, Mar. 1
  • Smooth maps between surfaces (p. 83)
    • The book gives the case when each surface is covered by a single surface patch; for the general definition, we consider a surface patch on the first surface which is mapped into a given surface patch on the second surface.
    • The notion of smoothness does not depend on which surface patch you use, because transition maps are smooth.
  • Diffeomorphisms of surfaces (p. 83)
  • Fact: smooth maps are continuous, hence diffeomorphisms are homeomorphisms.
  • Local diffeomorphisms (p. 83)
    • A plane wrapping around a cylinder is a local diffeomorphism (Example 4.3.2)
  • We define a map from a surface to Rn to be smooth if the map obtained by composing it with any regular surface patch is smooth. (Exercise 4.3.1)
  • Tangent vectors to a surface at a point, and the tangent space to the surface at a point (Definition 4.4.1)
Fri, Mar. 3 Midterm in class
Mon, Mar. 6
  • Every smooth curve on a surface is the composition of a smooth plane curve with a surface patch (p. 85)
    • This follows from the Inverse Function Theorem
  • Inverse Function Theorem (Theorem 5.6.1)
    • If a function F has an invertible derivative matrix DFp at a point p, then F restricts to a diffeomorphism between an open subset containing p and an open subset containing F(p)
  • First corollary: if σ:U→V is a surface patch, then for each p∈V, the inverse σ-1:V→U extends to a smooth map on an open subset of p.
    • That is, there is an open subset W⊂R3 containing p and a smooth map G:W→U with G(x)=σ-1(x) for x∈V.
  • Second corollary: the transition functions between regular surface patches are smooth (Proposition 4.2.6)
  • The tangent space to a surface at a point is the span of the partial derivatives of any surface patch (Proposition 4.4.2)
    • Proof next class
9 Wed, Mar. 8
  • Any map between surfaces which is a restriction of a smooth map on an open subset of R3 is smooth
    • More generally, it suffices for this to hold for the restriction of the function to an open neighbourhood of each point
    • The proof uses the Inverse Function Theorem
  • Any smooth, regular, injective map σ:U→S from U⊂R3 open to a surface S is automatically a regular surface patch
    • That is, it is not necessary to explicitly check the continuity of the inverse
    • This is also proven using the Inverse Function Theorem
  • Proof of Proposition 4.4.2 (stated last class)
  • Corollary: the tangent space to a surface at any point is a 2-dimensional linear subspace of R3 (i.e., a plane)
  • The derivative Dpf of a map f between surfaces at a point p (Definition 4.4.3)
  • Beginning of the proof that Dpf(v) doesn't depend on the chosen curve with tangent vector v
  • First lemma: the same description is valid for the derivative matrix DpF of a smooth function F:U→V between open subsets U⊂Rm and V⊂Rn
    • That is, if γ(0)=p and γ'(0)=v, then DpF(v)=(Fᐤγ)'(0)
    • The proof is immediate by the chain rule
Fri, Mar. 10
  • Proof that Dpf(v) doesn't depend on the chosen curve with tangent vector v
    • The proof used that the derivative Dxσ:R2→Tσ(x)S of a regular chart σ is linear bijection
  • The derivative of a map of surfaces at a point is a linear map (Proposition 4.4.4)
  • The derivative of the identity is the identity, the derivative of the composite is the composite of the derivatives, and the derivative of a diffeormopshism invertible (Proposition 4.4.5)
  • A map whose derivative at every point is invertible is a local diffeomorphism (Proposition 4.4.5)
    • Proof next class
Mon, Mar. 13 Spring recess. No class!
Wed, Mar. 15 Spring recess. No class!
Fri, Mar. 17 Spring recess. No class!
Mon, Mar. 20
  • Normal vector to a surface at a point (p. 89)
  • Standard unit normal of a surface patch (p. 89)
  • Orientable surfaces (Definition 4.5.1)
  • Oriented surface (p. 90)
  • A surface is orientable if and only if it can be made into an oriented surface (Proposition 4.5.2)
  • Convention: when considering an oriented surface, we will only consider those surface patches for which the standard unit normal agrees with the chosen orientation (p. 90)
  • The Möbius strip is not orientable (Example 4.3.5)
10 Wed, Mar. 22
  • Review of some linear algebra facts
    • Any linear subspace V⊂R3 has dimension 0, 1, 2, or 3
    • If V⊂V'⊂R3 are both linear subspaces, then dim V≤dim V', with equality if and only V=V'
    • The span of any non-zero vector is 1-dimensional
    • The span of any two independent vectors is 2-dimensional
    • (In particular, any two independent vectors in a tangent space to a surface span the whole tangent space.)
    • If w is orthogonal to both v and w, then it is orthogonal to Span(v,w)
    • If T:V→W is a linear map, then the image im(T)⊂W is a linear subspace
    • If T:V→W is a linear map and dim(V)=dim(W) then T is injective iff it's surjective iff it's bijective
  • Any set S (locally) defined by a smooth function f whose gradient is non-vanishing on S is a smooth surface (Theorem 5.1.1)
    • Also, the gradient f gives a normal vector to S at each point
  • The sphere and circular cone minus the origin (Examples 5.1.2-5.1.3)
  • Quadric surfaces (Definition 5.2.1)
    • Classification of quadric surfaces up to direct isometry (Theorem 5.2.2)
Fri, Mar. 24
  • Proof of Proposition 4.4.5, which was stated in class on Friday, March 10.
  • Ruled surfaces (Example 5.3.1)
  • Generalized cylinders (p. 105)
  • Generalized cones (p. 106)
  • Surfaces of revolution (Example 5.3.2)
  • Compact subsets of Rn (p. 109)
  • Planes, and open disks are not compact, but spheres and tori are (p. 110)
  • Informal definition of a compact surface of genus g
    • Every compact surface is diffeomorphic to the compact surface of genus g for some g (Theorem 5.4.4)
    • Corollary 5.4.5: every compact surface in R3 is orientable
Mon, Mar. 27
  • The first fundamental form (Definition 6.1.1)
  • Symmetric bilinear forms and inner products (Appendix 0)
    • Any symmetric bilinear form <-,-> is determined by the function q(v)=.
  • Computing lenghts of curves using the first fundamental form
  • First fundamental form of a plane (Example 6.1.2)
11 Wed, Mar. 29
  • "Substitution" interpretation of computing arc-lengths using first fundamental form
    • To compute the integral over γ of ∫(Edu2+2Fdudv+Gdv2)1/2, simply express u and v as functions of t, find the corresponding expressions for du and dv, and "substitute" into the integral
  • First fundamental form of a surface of revolution (Example 6.1.3)
    • Comptuation of the arc length of parallels and profile curves
  • First fundamental form of a generalied cylinder (Example 6.1.4)
  • First fundamental from of a generalized cone (Example 6.1.5)
Fri, Mar. 31
  • First fundamental form of a sphere (Example 6.1.3)
  • Local isometries (Definition 6.2.1)
  • A smooth map is a local isometry if and only if its dervative at each point is a linear isometry (Theorem 6.2.2)
  • Corollary: every local isometry is a local diffeomorphism (p. 128)
  • Corollary 6.2.3: a local diffeomorphism f:S1→S2 is a local isometry iff any surface patch σ for S1 has the same first fundamental form as the surface patch fᐤσ for S2
Mon, Apr. 3
  • The plane and cylinder are locally isometric (Example 6.2.4)
    • Since there is a smooth map between them taking surface patches to surface patches with the same first fundamental form
  • Expressing angles between tangent vectors in terms of the first fundamental form (p. 133)
  • Normal and geodesic curvatures (Definition 7.3.1 and Proposition 7.3.2)
  • The second fundamental form (p. 160)
12 Wed, Apr. 5
  • Second fundamental form of a plane, generalized cylinder, and sphere (Examples 7.1.1 and 7.1.2)
  • The Gauss map (p. 162)
  • The Weingarten map (Definition 7.2.1)
  • The second definition of the second fundamental form (p. 163)
    • This agrees with the first definition (next class)
Fri, Apr. 7
  • The second definition of the second fundamental form (using the Weingarten map) is symmetric
    • We used: a bilinear form is symmetric as soon as it is symmetric on all the vectors in a given basis
  • The two definitions of second fundamental form agree (Proposition 7.3.3)
  • Normal sections (p. 169)
  • The curvature of a normal section at a point agrees with its normal curvature (Corollary 7.3.5)
Mon, Apr. 10
  • Gaussian and mean curvature (Definition 8.1.1)
  • Linear algebra review
    • Representation of a linear operator (that is, a linear map from a vector space to itself) as a matrix with respect to a basis
    • Determinant and trace of a linear operator
  • Matrix representation of the Weingarten map in a surface patch (Proposition 8.1.2)
  • Gaussian curvature of a surface of revolution (Example 8.1.4)
    • A plane has 0 Gaussian curvature
    • A cylinder also has 0 Gaussian curvature
    • A sphere has constant Gaussian curvature 1
13 Wed, Apr. 12
  • Rules surfaces have non-positive Gaussian curvature (Example 8.1.5)
  • Gaussian curvature is rate of change of area of Gauss map (Theorem 8.1.6)
  • Applications of Theorem 8.1.6 (Example 8.1.7)
    • The plane and cylinder have 0 Gaussian curvature
    • The unit sphere has constant Gaussian curvature 1
    • The sphere of radius R has constant Gaussian curvature 1/R
  • Principal curvatures and principal vectors (p. 187)
    • Linear algebra review: self-adjoint operators are diagonalizable, and eigenvectors for different eigenvalues are orthogonal (Appendix A)
  • Gaussian and mean curvatures are the product and average of the principal curvatures (Proposition 8.2.3)
  • Next time: the principal curvatures at a point are the maximal and minimal normal curvatures of curves at that point
Fri, Apr. 14
  • Umbilic points (p. 187)
  • Every tangent space has an orthonormal basis of principal vectors (Corrolary 8.2.2)
  • The principal curvatures are the maximal and minimal normal curvatures (Corrolary 8.2.5)
  • Review of characteristic polynomial
  • Finding the principal curvatures and vectors in a surface patch (Proposition 8.2.6)
    • Doing this for the sphere (Example 8.2.7) and cylinder (Example 8.2.8)
  • If every point on a connceted surface is umbilic, it is (an open subset of) a sphere or a plane
Mon, Apr. 17
  • Geometric intepretation of second fundamental form (p. 193)
    • Elliptic, hyperbolic, parabolic, and planar points
    • Example 8.2.11: curvature of the torus
  • Geodesics (Definition 9.1.1)
  • Geodesics have constant speed (Proposition 9.1.2)
  • A unit speed curve is geodesic if and only if its geodesic curvature vanishes (Proposition 9.1.3)
  • Straight lines are geodesics (Proposition 9.1.4)
  • Great circles on spheres and parallels on generalized cylidners are geodesics (Examples 9.1.7,9.1.8)
14 Wed, Apr. 19
  • The geodesic equations (Proposition 9.2.3)
  • The Christoffel symbols (p. 172)
  • Existence and uniqueness of geodesics with prescribed point and tangent vector (Proposition 9.2.4)
  • Classification of geodesics on planes and spheres (Examples 9.2.5-9.2.6)
  • Geodesics are preserved by local isometries (Corollary 9.2.7)
  • The Christoffel symbols only depend on the first fundamental form (Proposition 7.4.4)
Fri, Apr. 21
  • Classification of geodesics on the cylinder (Example 9.2.8)
  • Variations of curves (p. 236)
  • A unit-speed curve is a geodesic if and only if it is a critical point of the length functional (Theorem 9.4.1)
    • In particular, any path of shortest length between two points is a geodesic
Mon, Apr. 24
  • Geodesics are locally shortest paths (i.e., for any two sufficiently close points, any geodesic connecting them is the shortest path between them)
    • The proof uses the Gauss lemma: the geodesics through a point are a perpendicular to small concentric circles around that point.
  • Statement of Theorema Egregium (Theorem 10.2.1)
  • Corollary: any map of part of the Earth must distort distances
  • Explanation of why we fold pizza
  • Statement of Gauss-Bonnet (Theorem 13.4.5)
  • Beginning of proof of Theorema Egregium: the Gauss equations (Proposition 10.1.2)
15 Wed, Apr. 26
  • Rest of proof of Theorema Egregium
  • Beinning of proof Gauss-Bonnet for simple closed curves (Theorem 13.1.2)
Fri, Apr. 28
  • Rest of proof of Gauss-Bonnet for simple closed curves
  • Gauss-Bonnet for curvilinear polygons (Theorem 13.2.2)
  • Triangulations and Euler characteristic (§13.4)
  • Proof of Gauss-Bonnet (Theorem 13.4.5)