Analysis II: 18.101 (Fall 2002)

Lectures

Textbooks

Grader

Examinations and Homework

• HW #1, due September 17: Spivak 2-7, 2-12, 2-13, 2-14, 2-15, 2-24, 2-32, 2-34, 2-35; Munkres 5-2, 5-4, 7-1, 7-2, 7-3.

• HW #2, due September 24: Munkres 8-1, 8-2, 8-3, 8-4, 8-5, 9-1, 9-2, 9-5, 9-6; Spivak 2-36, 2-37, 2-38, 2-39.

• HW #3, due October 1: Munkres 10-3, 11-1, 11-2, 11-3, 11-4, 11-6, 11-8; Spivak 3-1, 3-2, 3-7, 1-30, 3-12, 3-14, 3-18.

• HW #4, due October 8: Munkres 12-2, 12-3(b), 12-4, 13-1, 13-2, 13-4; Spivak 3-26, 3-27, 1-17, 1-18, 3-30.

• HW #5, due October 22: Munkres 15-1, 15-2, 15-4, 16-1, 19-6, 20-1, 20-2, 20-3, 20-4; Spivak 3-36, 3-38, 3-41.

• HW #6, due October 31: Spivak 4-1, 2, 3, 9, 13, 14, 18, 19, 20, 21.

• HW #7, due November 12: Spivak 4-23, 26, 28, 29, 30; 5-4, 6, 7 (assume n=3).

• HW #8, due November 19: Spivak 5-9, 13, 14, 15.

• HW #9, due December 5: Munkres 37-4, 37-5, 37-6; Spivak 5-20, 22, 23, 25, 26, 29, 31, 34, 35.

Brief Lecture notes

• Lecture 1: September 5
  
  1. Differentiability of mappings of Euclidean spaces.
  2. Partial derivatives.
  3. Continutuity at a point of differentiability.
  4. Differentiability of components
  5. Jacobian matrix.
• Lecture 2: September 10
  
  1. Continuous differentiability -- continuity of partial derivatives.
  2. Continuous differentiability implies differentiability.
  3. Chain rule.
• Lecture 3: September 12
  
  1. Hessian and higher order derivatives
  2. Symmetry of Hessian of C^2 functions.
  3. Mean value theorem.
• Lecture 4: September 17
  
  1. Taylor's Theorem
  2. Maximum Principle.
  3. Inverse function theorem.
• Lecture 5: September 19
  
  1. Inverse function theorem, smoothness of the inverse.
  2. Implicit funtion theorem.
• Lecture 6: September 24
  
  1. The Riemann integral.
  2. Integral of bounded functions on rectangles.
  3. Riemann's condition.
• Lecture 7: September 26
  
  1. Which function are integrable?
  2. Measure zero sets.
  3. Continuous almost everywhere implies integrable
• Lecture 8: October 1
  
  1. Integrable implies continuous almost everywhere.
  2. Integral of bounded functions on bounded sets.
  3. Properties of the integral (linearity, comparison, additivity).
  4. Problems with the Riemann integral.
• Lecture 9: October 3
  
  1. Fubini's Theorem.
  2. Simple regions.
• Lecture 10: October 8
  
  1. Partitions of unity
  2. Extended integral - integral of locally bounded functions on open sets (including unbounded open sets).
• Lecture 11: October 10
  
  1. Change of variable formula.
• Lecture 12: October 17
  
  1. Change of variable formula continued.
  2. Sard's Theorem.
  3. Orientation and isometries.
• Lecture 13: October 22
  
  1. Multilinear algebra; tensors.
  2. Alternating tensors and wedge product.
• Lecture 14: October 24
  
  1. Vector fields and differential forms.
  2. Pull-back of forms.
• Lecture 15: October 29
  
  1. The d operator.
  2. Closed and exact forms.
  3. Poincare Lemma.
• Lecture 16: October 31
  
  1. Chains and the boundary operator.
  2. Integration of differential forms.
  3. Stokes' Theorem.
• Lecture 17: November 5
  
  1. Submanifolds of R^n.
  2. Local coordinates on submanifolds.
• Lecture 18: November 7
  
  1. Submanifolds with boundary.
  2. Vector fields and differential forms on submanifolds.
  3. Orientations.
• Lecture 19: November 12
  
  1. Orientable manifolds.
• November 14
  
  1. Exam.
• Lecture 20: November 19
  
  1. Solutions to exam.
• Lecture 21: November 21
  
  1. Integration on manifolds.
  2. Stokes' Theorem on Manifolds.
• Lecture 22: November 26
  
  1. Volume Elements.
  2. Divergence Theorem.
  3. Classical Stokes' Theorem.
• Lecture 23: December 3
  
  1. Manifolds.
• Lecture 24: December 5
  
  1. Tangent bundle, vector fields, differential forms.
  2. Stokes' Theorem for manifolds.
  3. Riemannian metrics.
• Lecture 25: December 10
  
  1. De Rham cohomology.