Calculus-on-Manifolds-S18

Materials, Resources and Accommodating Disabilities:

Textbook:

Copies of Calculus on Manifolds by Spivak will be distributed for students to use for one semester.

It is also recommended to have a calculus book:

Larson, Hostetler and Edwards, Calculus (Early Transcendentals), Houghton Mifflin,

Other good books you might wish to purchase:

A First Course in Geometric Topology and Differential Geometry by Bloch

Online Resources:

Wikipedia has articles on most topics in this course with links to further reading.

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 Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.

Course Objectives:

At the end of the course students should know:

1. Inverse and implicit function theorems, 2. Manifolds. 3. Differential Forms. 4. Fubini's Theorem. 5. Integration on Chains. 6. Stokes’ and Green's theorems. 7. Introduction to Riemannian geometry

These objectives will be assessed on the final exam along with other important techniques.

Topics:

• Basic Theory of Metric Spaces:

• Quantifiers, Sets, Unions and Intersections, Balls and Planes in Euclidean space

• Normed Vector Spaces, Inner Product Spaces and Tangent Spaces

• Balls and Open sets in Euclidean Space and Metric Spaces

• Project 1 on Basic Theory

• Continuity:

• Review of Linear Maps on Vector Spaces: Images, Null Spaces, Inverses, and Rank

• Continuous Maps on Euclidean Space and Metric Spaces

• Images and Preimages of Continuous Maps on Metric Spaces

• Injective and Surjective Maps, Homeomorphisms and Inverses

• Project 2 on Continuity

• Differentiation:

• Review of Smooth Maps on Euclidean Space

• Tangent Vectors, Gradients, Jacobian Matrix,

• Chain Rule and Product Rule

• Inverse Function Theorem and Diffeomorphisms

• Implicit Function and Rank Theorem

• Manifolds: Metric Spaces with Smooth Charts (Sphere, Torus)

• Project 3 on Differentiation

• Integration:

• Measure Zero

• Integrable Functions and Fubini's Theorem

• Partitions of Unity

• Change of Variables dxdy to rdrdtheta

• Tensor Products and Wedge Products

• Differential Forms and Determinants w to dw

• Boundaries and Stoke's Theorem

• Project 4 on Integration

• Manifolds:

• Fields and Forms on Manifolds

• Stoke's Theorem on Manifolds

• Green's Theorem and Divergence Theorem

• Project 5 on Manifolds

• Final Exam on All Topics (Projects 3-6 and Riemannian Manifolds)

Schedule: (this will be updated daily including the homework assignments)

Lesson 1 (1/29) Euclidean Space: vectors, inner products, planes, balls

email me for the notes and problems for Lesson 1

Problem Session (1/31 at 7pm or 9pm):

see fourth page of notes for Lesson 1

Lesson 2 (1/31) Vector Spaces, Inner Product Spaces, Normed Spaces

2/2 is Mathfest at the CUNY Graduate

email me for the notes and problems for Lesson 2

Problem Session (2/5 at 7pm or 9pm):

see problems in notes for Lesson 2

Lesson 3 (2/5) Tangent Spaces, Tangent Planes and Metric Tensors

email me for the notes and problems for Lesson 3

Problem Session (2/7 at 7pm or 9pm):

see problems in notes for Lesson 3

Lesson 4 (2/7) Metric Spaces: Balls and Open Sets

email me for the notes and problems for Lesson 4

Lehman is closed on 2/12 (students can meet)

Problem Session (2/14 at 7 or 9 pm)

see problems in notes for Lesson 4

Project I on Basic Theory is the following 10 problems due by email by midnight on 2/19:

Lesson 2: submit 1, 3, and 4;

Lesson 3: submit 3a, 3b, and 3c;

Lesson 4: submit 1, 2, 5, and 9;

Lesson 5 (2/14) Continuous Maps on Euclidean Space and Metric Spaces

email me for the notes and problems for Lesson 5

Lehman closed on 2/19 (students can meet. project is due at midnight)

Problem Session (2/20 at 7 or 9 pm)

see problems in notes for Lesson 5

Lesson 6 (Tuesday 2/20) Images and Preimages of Continuous Maps on Metric Spaces

email me for the notes and problems for Lesson 6

Problem Session (2/21 at 7 or 9 pm)

see problems in notes for Lesson 6

Lesson 7 (2/21: sub) Review of Linear Maps on Vector Spaces: Images, Null Spaces, Inverses, and Rank

email me for the notes and problems for Lesson 7

Problem Session (2/26 at 7 or 9 pm)

see problems in notes for Lesson 7

Lesson 8 (2/26) Injective and Surjective Maps, Homeomorphisms and Inverses

email me for the notes and problems for Lesson 8

Problem Session (2/28 at 7 or 9 pm)

see problems in notes for Lesson 8

Project 2 on Continuity is due at midnight by email on Sunday 3/4 (may submit by midnight on Saturday 3/10)

Lesson 5: HW1abc; Lesson 6: HW9, HW10; Lesson 7: HW3; Lesson 8: HW2, HW3, HW4, HW5, HW6;

Lesson 9 (2/28) Review of Smooth Maps on Euclidean Space, Tangent Vectors, Gradients, Jacobian Matrix,

email me for the notes and problems for Lesson 9

Problem Session (3/5 at 7 or 9 pm)

see problems in notes for Lesson 9

Lesson 10 (3/5) Jacobean Matrix and Chain Rule

email me for the notes and problems for Lesson 9

Problem Session (moved to 3/12 at 7 or 9 pm)

see problems in notes for Lesson 10

Lesson 11: (3/7) snow day (will schedule an extra lesson later)

Lesson 12 (3/12) Inverse Function Theorem and Diffeomorphisms

email me for the notes and problems for Lesson 11

Problem Session (3/14 at 7 or 9 pm)

see problems in notes for Lesson 11

Lesson 13 (3/14) Implicit Function Theorem

email me for the notes and problems for Lesson 12

Problem Session (3/19 at 7 or 9 pm)

see problems in notes for Lesson 12

Lesson 14 (3/19) Manifolds: Metric Spaces with Smooth Charts (Sphere, Torus)

email me for the notes and problems for Lesson 13

Project 3 on Differentiation is due 4/8 at midnight

do the problems listed on the Lesson 15 review for the Midterm

Lesson 15 (3/21) Review (snow day: will schedule an extra lesson later)

one page of problems to review was emailed to the class

Lesson 16 (3/26 sub) Midterm

Lessons 17 (3/28) Integration

email me for the notes and problems for Lesson 17

Problem Session (4/9 at 7 or 9 pm)

see problems in notes for Lesson 17

Spring Recess

Lesson 18 (4/9): Measure Zero

email me for the notes and problems for Lesson 18

Problem Session (4/16 at 7 or 9 pm)

see problems in notes for Lesson 18

Lehman has Friday classes on 4/11 but students can meet in 137

We will have a Makeup Lesson on 4/11 reviewing Exam I

Lesson 19 (4/16) Integrable Functions and Fubini's Theorem

email me for the notes and problems for Lesson 19

Problem Session (4/18 at 7 or 9 pm)

see problems in notes for Lesson 19

Lesson 20 (4/18) Partitions of Unity

email me for the notes and problems for Lesson 20

Problem Session (4/23 at 7 or 9 pm)

see problems in notes for Lesson 20

Lesson 21 (4/23) Integration over Open Sets and Change of Variables

email me for the notes and problems for Lesson 21

Problem Session (4/25 at 7 or 9 pm)

see problems in notes for Lesson 21

Lesson 22 (4/25) Tensors, Tensor Products, and Wedge Products

email me for the notes and problems for Lesson 22

Problem Session (4/30 at 7 or 9 pm)

see problems in notes for Lesson 22

Lesson 23 (4/30) Differential Forms and Determinants w to dw

email me for the notes and problems for Lesson 23

Problem Session (5/2 at 7 or 9 pm)

see problems in notes for Lesson 23

Lesson 24 (5/2) Problem Session instead of the lecture

Problem Session (5/7 at 7 or 9 pm)

see problems in notes for previous lessons

• Project 4 on Integration is due 5/5:

• Lesson 17 (3) Lesson 18 (4) Lesson 19 (1) Lesson 20 (1) (7) Lesson 21 (1) (2) (4) (5) (7)

Lesson 25 (5/7) Integrating Differential Forms

email me for the notes and problems for Lesson 25

Problem Session (5/9 at 7 or 9 pm)

see problems in notes for Lesson 25

Lesson 26 (5/9) Stoke's Theorem on Manifolds

email me for the notes and problems for Lesson 26

Problem Session (5/14 at 7 or 9 pm)

see problems in notes for Lesson 26

• Project 5 on Manifolds will be due May 26 at 11:59 pm

• Lesson 22: 1, 2, a couple parts of 5 and of 7 (but any part might be on the final)

  • Lesson 23: 0, 1, 2, 3, 4, 5, 6, 7, 8 (any could be on the final)

  • Lesson 25: 1, 2, 5, 9, 10, 11, 12 (11 or 12 could be on the final)

  • Lesson 26: 2, 3. Lesson 27: 3 Lesson 28: all problems (Lessons 26-8 not on final)

Lesson 27 (5/14) Divergence Theorem and Green’s Theorem

email me for the notes and problems for Lesson 27

Problem Session (2/28 at 7 or 9 pm)

see problems in notes for Lesson 27

Lesson 28 (5/16) Review for the Final

Final Exam during Finals Week: May 23 8-10 pm Gillet 333

(two hour exam: but you may start at 7 pm outside 333 if you wish and take a 30 min break at sun down or start late and end two hours later)

Description of the Final: (covers the second half of the semester)

You may use notes, phones and books during the final. Even google what you wish. It is helpful to have a reference sheet of key formulas or a guide to course notes. You may not text or talk or email anyone during the final.

(1) 20% Manifolds: find the transition map for a pair of given charts and verify the transition maps are diffeomorphisms and in same oriented atlas (by checking det of the derivative is positive) See Lesson 13 and Lesson 25

(2) 10% Riemann Sums: something like: (What is the upper Riemann sum for f(x,y) with an even partition of n x m rectangles in [1,3]x[2,5]? ) See Lessons 17 and 19

(3) 10% Measure Zero: proof

(prove a given set has measure 0, like hw) See Lesson 18

(4) 10% Partition of unity two short questions of the following style: (give an example of a C infty function which is 0 for x le 0 but positive elsewhere) (verify a given function is C^1 using difference quotient) (which of the following is an open cover) (state the defn of partition of unity) (which of the following is a partition of unity) See Lesson 20

(5) 10% Volume: compute doing the integration out (compute like hw in Lesson 21)

(6) 10% Tensor and wedge products: a linear algebra style proof (any part of Lesson 22 hw 5 or 7 might be assigned on the final),

(7) 15% Differential forms: three short questions (anything like Lesson 23 hw 0, 1, 2, 3, 4, 5, 6, 7, 8 could be on the final)

(8) 10% Integration on manifolds: deriving formulas (something like Lesson 25 hw 11 or 12 could be on the final)

(9) 5% Stoke’s Thm (apply it to a given chain and form). (Lessons 26-28)