MATH 134 CALCULUS ON MANIFOLDS Class times and location: 12 MWF location TBA Office Hours: Mondays 1-2pm, Tuesdays 4-5pm, also by appointment. Summary: Generalization of multivariable calculus to the setting of manifolds in real n-space, as used in the study of global analysis and geometry. Differentiable mappings of linear spaces, the inverse and implicit function theorems, differential forms, integration on manifolds, the general version of Stokes's theorem, integral geometry, applications. Prerequisite: Mathematics 21a,b and familiarity with proofs as in Mathematics 101, 112, 121, or the equivalent. Texts: Calculus on Manifolds by M. Spivak; Differential Topology by V. Guillemin and A. Pollack. Examination policy: 2 midterm exams and 1 final exam. Homework policy: weekly worksheets. It is OK to discuss the problems amongst yourselves. However each student must hand in their own solutions that they have written themselves. (Copying someone else's homework is unacceptable.) To make the job of grading easier, could you please follow the following guidelines for homework: Write your name on your HW. Neat, legible handwriting. I will not grade anything I cannot read!!! Write on ONE side of the paper only. The problems should be in the order assigned. Staple (or paper-clip) all pages together. LATE HW: I will not grade late HW, unless you have a good reason (ex illness). (If you are sick, let me know and we can meet and sort out HW extensions and some extra help so you can catch up on the work.) If you can't make it to the class when the HW is due, you have several options: 1) Give your HW to a friend to hand in. 2) Put it in my mail box in the Math Department (Level 3 Science Center). 3) Slide it under my office door (535 Science Center). Grading policy: worksheets 1/5; midterms 2/5; final exam 2/5. Attendance: Attendance will not be taken at each class. However, it is much harder to learn the material on your own, so you are strongly encouraged to attend each class. You must attend each of the 2 midterm exams and final exam. Make-up exams will only be given in special circumstances. Drop Date: The drop date for the course is Monday October 18. Final exam date: Tuesday January 25. Syllabus (subject to final revision): Examples of manifolds, definition of a differentiable and topological manifold Orientation and Jacobian matrix Solutions of constraint equations and the Implicit and Inverse Function Theorems Tangent vectors, tangent spaces Diffeomorphism between manifolds Maps on Tangent spaces Regular values and the Preimage Theorem Vector fields and flows Dual space and differential of a function Midterm 1 somewhere around here Tensors and exterior algebra Differential forms Singular chains Fundamental Theorem of Calculus Exterior Derivative Poincare Lemma Stokes Theorem Classical Theorems Midterm 2 somewhere around here Extra topics including: Lie derivatives Riemannian metrics Covariant Derivatives and Connections Geodesic equations Applications to mechanical systems