Spring 2019 MAT 322: Analysis in several dimensions | ||

Schedule | MW 2:30-3:50 Physics P117 | |

Instructor | Robert Hough | |

Office hours | TuTh 9-10am in Math Tower 4-118, W 6-7pm in Math Learning Center | |

Grader | Jin-Cheng Guu | |

Description | Continuity, differentiation, and integration in Euclidean n-space. Differentiable maps. Implicit and inverse function theorems. Differential forms and the general Stokes' theorem. | |

Prerequisites | C or higher in MAT 203, MAT 220, MAT 307, or AMS 261; C or higher in MAT 310 or MAT 315; B or higher in MAT 320 | |

Textbook | James Munkres. Analysis on manifolds. CRC Press (2018). | |

Homework | Weekly problem sets will be assigned, and collected in class on Monday. Late homework is not accepted, but under documented extenuating circumstances the grade may be dropped. Your lowest homework grade will be dropped at the end of the class. | |

Grading | Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%. |

** Syllabus/schedule (subject to change)**

Mon 1/28 | 1. | Linear algebra, matrices | Read pages 1-40 |

Wed 1/30 | 2. | Topology in Euclidean space | HW due 2/4: p.9 #1, 2, 4, p.24 #1, 6, p.30 #2, 7, 8, p.39 #1, 3 |

Mon 2/4 | 3. | The derivative | Read pages 41-55 |

Wed 2/6 | 4. | Continuously differentiable functions | HW due 2/11: p.48 #1, 2, 4, p.54 #1, 2, 4, 5, 8, 9, 10 |

Mon 2/11 | 5. | The chain rule | Read pages 56-62 |

Wed 2/13 | 6. | The inverse function theorem | HW due 2/18: p.63 #1, 2, 3, p.70 #1, 2, 3, 4, 5 |

Mon 2/18 | 7. | The implicit function theorem | Read pages 63-90 |

Wed 2/20 | 8. | The integral over a rectangle | HW due 2/25: p.78 #1, 2, 3, 4, 5, 6, p.90 #1, 3, 6, 7 |

Mon 2/25 | 9. | The existence and evaluation of the integral | Read pages 91-134 |

Wed 2/27 | 10. | The integral over a bounded set, improper integrals | HW due 3/6: p.97 #1, 2, 4, 5, 10, p.103 #3, 4, p.111 #4, 5, |

Mon 3/4 | Midterm 1 Practice Problems, Midterm solutions | Read pages 135-143 | |

Wed 3/6 | 11. | Partitions of unity | HW due 3/11: p.120 #1, 2, 6, p.132 #4, p.143 #1, 2, 3 |

Mon 3/11 | 12. | The change of variables formula, diffeomorphisms | Read pages 144-178 |

Wed 3/13 | 13. | Change of variables and applications | HW due 3/25: p.151 #4,5, p.160 #1, 3, 4, p.167 #1, 2, 4, 6, p.177 #1 |

Mon 3/18 | No class - Spring Break | ||

Wed 3/20 | No class - Spring Break | ||

Mon 3/25 | 14. | The volume of a parallelopiped, sub-manifolds of Euclidean space | Read pages 179-218 |

Wed 3/27 | 15. | The boundary of a manifold, integration on manifolds | HW due 4/1: p.187 #5, p.202 #3, 4, 5, 6, p.208 #2, 5, 6, p.218 #3, 5, 7 |

Mon 4/1 | 16. | Multilinear algebra | Read pages 219-235 |

Wed 4/3 | 17. | Alternating tensors | HW due 4/10: p.226 #1, 2, 4, 6, 8, p.236 #1, 2, 3, 5 |

Mon 4/8 | 18. | The wedge product | Read pages 236-251 |

Wed 4/10 | 19. | Tangent vectors and differential forms | HW due 4/15: p.243 #2, 3, 4, 5, 6, p.251 #1, 2, 3, 4, 5 |

Mon 4/15 | 20. | Review | |

Wed 4/17 | Midterm 2 , Practice Problems, Solutions to Practice ProblemsSolutions to Midterm 2 | ||

Mon 4/22 | 21. | The differential operator, applications | Read pages 252-274 |

Wed 4/24 | 22. | Differentiable maps | HW due 4/29: p.260 #2, 4, 5, 6, 7, p.265 #5, p.273 #2, 3, 5 |

Mon 4/29 | 23. | Integrating forms | Read pages 275-296 |

Wed 5/1 | 24. | Orientable manifolds | HW due 5/6: p.280 #1, 2, 3, 4, p.291 #1, 3, 4, 8, p.296 #3 |

Mon 5/6 | 25. | Generalized Stokes' Theorem | Read pages 297-322 |

Wed 5/8 | 26. | Applications | HW due 5/13: p.308 #2, 3, 4, 6, p.320 #1, 2, 4 |

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