MA410 Differential Geometry


Instructor: Nikolay S. Strigul
E-mails: nstrigul@stevens.edu
Office Hours: by appointment

Lectures: MW, 02:00-03:15 pm, McLean 414
Homeworks: There will be weekly homework assignments.
Exams: There will be Midterm and Final Exams.
Quizzes: There will be regular quizzes covering definitions and theorems.

Grading:

Course program Course announcement (PDF) Course program (PDF)

General comments:

MA540 introduces to students major ideas of differential geometry and its applications to physics. Upon completion of this course students will have knowledge of the geometry of curves and surfaces, understand how calculus, topology and linear algebra contribute to studying geometrical objects, will be able to solve typical problems associated with this theory, and will be able to use standard software to visualize curves and surfaces and to perform standard calculations.

Textbooks

1) Do Carmo, Manfredo, Differential Geometry of Curves and Surfaces, Prentice Hall.
2) O'Neill, Barrett, Elementary Differential Geometry
3) Gray, Alfred, Elsa Abbena, and Simon Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica.
4) Zorich, Vladimir, Mathematical Analysis
5) Dugundji, James, Topology.


Course program:

Lecture 1. - Vector spaces

1. Field
2. Vector space
3. Linear dependence and independence
4. Basis and dimension of a vector space
5. Finite and infinite dimensional vector spaces
6. Linear mapings and forms. Rank of a linear mapping
7. Isomorphisms

Lecture 2. - n-dimensional Euclidean spaces

8. Norm
9. Scalar product. Orthogonally
10. Vector product
11. Direct sum of vector spaces over the same field
12. Euclidean spaces


Lecture 3. - Elements of topology 1

13. Topological spaces
14. Open and closed sets
15. Injective, subjective and bijective functions

Lecture 4. - Elements of topology 2

16. Open and closed functions
17. Continuous functions
18. Homeomorphism and diffeomorphism.

Lecture 5. - Curves in n-dimensional Euclidean space

19. Curves and their visualization in Mathematica
21. Regular curves
22. Vector field along the curve
23. The length of a curve
24. Parameterization of a curve by arc length

Lecture 6. - Frenet frame

25. Tangent, normal and binormal vectors
26. Curvature, curvature radius
27. Torsion
28. Applications to the motion of particles in space
29. Computer simulations

Lecture 7. -Review and exersises

Lecture 8. -Surfaces in n-dimensional Euclidean space 1

30. Tangent vectors to n-dimensional Euclidean space
31. k-dimensional surface
32. Local chart
33. Atlas of a surface

Lecture 9. -Surfaces in n-dimensional Euclidean space 2

30. Smooth k-dimensional surface
31. Surface parametrization
32. Computer visualization in Mathematica

Lecture 10. - Orientation 1

33. Oriented Euclidean space
34. Orienting atlas of a surface
35. Orientable and non-orientable surfaces

Lecture 11. - Orientation 2

36. Boundary of a surface and its orientation

Lecture 12. - Area of a surface in Euclidean space

37. An area of a k-dimensional smooth surface
38. Applications
39. Computer simulations

Lecture 13. -Review and exersises

Lecture 14. - Midterm exam

Lecture 15. - Elements of multilinear algebra

40. Bilinear mapping
41. Alternating multilinear mapping
42. Tensor product of vector spaces
43. External product of vector spaces

Lecture 16. - Differential forms 1.

44. Real-valued differential p-form
45. Differential form defined on a smooth surface
46. Exterior differentiation of a differential form
47. Second fundamental form

Lecture 17. - Differential forms 2.

48. Work form of a vector field
49. Flux form of a vector field
50. Coordinate expressions of differential forms

Lecture 18. - Integral of differential form

51. k-form over a given chart of a smooth k-dimensional surface
52. Volume element. Integral of a form over an oriented surface
53. Applications: work of a field and a flux across the surface
54. Mass of a surface with given density

Lecture 19. -Review and exersises

Lecture 20. -The Gauss-Ostrogradskii formula

55. Green's Theorem
56. The Gauss-Ostrogradskii formula
57. Applications

Lecture 21. -Stokes' theorem

58. Stokes' theorem
59. Applications to vector analysis
60. Operators: grad, curl, div, and nabla

Lecture 22. -Gauss's Theorem Egregium

61. Rigid motions
62. Theorem Egregium

Lecture 23. -The Gauss-Bonnet theorem for surfaces

63. Degrees of maps between compact surfaces
64. The Gauss-Bonnet theorem

Lecture 24. -Manifolds 1

65. Separatiom axioms
66. Definition of a manifold
67. Smooth manifolds
68. Orientation of a manifold

Lecture 25. -Manifolds 2

69. Differential forms on manifolds

Lecture 26. -Review and exersises

Final exam.

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