1. Week 1 (1 Apr - 5 Apr 2013): Read 1-34. Do 1.7, 1.10, 1.22, 1.30, 2.4, 2.5, 2.7.
2. Week 2 (8 Apr - 12 Apr 2013): Read 34-45. Do 2.12, 2.13., 2.21, 2.22, 2.23, 2.24, 2-25, 2-26. Also Extra problem
3. Week 3 (15 Apr - 19 Apr 2013): Read 46-56. Do 2.29, 2.30, 2.31, 2.32, 2.35, 2.36, 2.37 [not part b], 2.38, 2.39
4. Week 4 (22 Apr - 28 Apr 2013): Read 56-73. Do 3.1-3.10.
5. Week 5 (29 Apr - 3 May 2013): Read 56-73. Do 1.18, 3.11, 3.12, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19. Exam Wednesday May 01 2013
6. Week 6 (6 May - 10 May 2013): Do 3.13, 3-20, 3.21, 3.22, 3.23, 3.26, 3.28, 3.29, 3.36.
7. Week 7 (13 May - 17 May 2013): Do 1.17, 3.30, 3.31, 3.32, 3.33, 3.34, 3.37, 3.38
1. Problem A1:  Prove or disprove the following assertion: ``Let U be a bounded open subset of R-n. Then the characteristic function of U is integrable in the extended sense over U.
2. Problem A2:   Prove or disprove the following assertion: ``If U is any unbounded open subset of R-n, then the characteristic function of U is not integrable in the extended sense over U.''
8. Week 8 (20 May - 24 May 2013): Assignment #8 Improper integrala
9. Week 9 (25 May - 30 May 2013): Assignment #9 Greens, Gausss and Stokes theorem
Memorial day
10. Week 10 (03 Jun - 07 Jun 2013): TBA

1. Improper Integrals
2. Change of variable Theorem
3. Green's, Gauss's, Stokes Theorem

Course Goals

1. In the first piece, we discuss the notion of differentiation in the multi-variate context. The derivative is presented as the best linear approximation; it is then related to the Jacobian matrix of partial derivatives and to directional derivatives. The chain rule is established and the elementary properties of the derivative are derived. The inverse and implicit functions are established.
2. We then turn to the theory of Riemann integration. The notion of upper and lower sums are introduced in the multi-variate context building on the theory introduced in Math 4/513 and Math 4/514. Sets of content zero and measure zero are introduced and it is shown that a function is Riemann integrable if and only if it is bounded and continuous except on a set of measure zero. Jordan measurability is defined in this context and Fubini's theorem is established (interchanging the order of integration is a bit complicated). Partitions of unity and compact exhaustions are presented and applied to discuss improper integrals; lecture notes are available to supplement Spivak's treatment. We conclude this section with the change of variables theorem; again, lecture notes are available to supplement Spivak's treatment. The change of variables theorem and Fubini's theorem is explored at length in Homework Assignment 8.
3. The final section of the course deals with the generalized Stoke's theorem and differential forms. At this point Spivak's treatment is just a bit too abstract and requires algebraic sophistication not really appropriate to the level of this course. Course notes are available. The exterior algebra will be presented in terms of universal properties, but the properties are stressed not the algebraic foundations. The role of the determinant in relation to top degree forms is discussed as is the change of variables and exterior differentiation in the discussion of the integration theory of differential forms. The generalized Stokes theorem is proved. The relationship of this theorem to the classical Green's, Gauss's, and Stoke's theorem is explored at length in homework assignment 9. If time permits, the fundamental theorem of algebra, the Brauer fixed point theorem (any continuous map from the disk to itself has a fixed point), and the Billiard ball theorem are presented as applications of this material.

Mathematics Department Undergraduate Grading Standards November 2011

• (1) Mathematics is hierarchical. A student who is given a grade of C or higher in a course must have mastery of that material that allows the possibility of succeeding in courses for which that course is a prerequisite.
• (2) Some mathematics courses are primarily concerned with techniques and applications. In such courses student success is measured by the student's ability to model , successfully apply the relevant technique, and bring the calculation to a correct conclusion. The department's 100-level courses and most calculus courses are examples in this category although these are not the only examples. Other courses such as this one are primarily concerned with theoretical structures and proof. In such courses student success is measured by the student's ability to apply the theorems and definitions in the subject, and to create proofs on his or her own using the models and ideas taught during the course. Many courses are partly hybrids incorporating both techniques and applications, and some element of theory. Some lean more toward applications, others more toward theory.

• A: Consistently chooses appropriate models, uses correct techniques, and carries calculations through to a correct answer. Able to estimate error when appropriate, and able to recognize conditions needed to apply models as appropriate.
• B: Usually chooses appropriate models and uses correct techniques, and makes few calculational errors. Able to estimate error when prompted, and able to recognize conditions needed to apply models when prompted.
• C: Makes calculations correctly or substantially correctly, but requires guidance on choosing models and technique. Able to estimate error when prompted and able to recognize conditions needed to apply models when prompted.
• D: Makes calculations correctly or substantially correctly, but unable to do modeling.
• F: Can neither choose appropriate models, or techniques, nor carry through calculations.

Rubric for pure courses:

• A: Applies the important theorems from the course. Constructs counterexamples when hypotheses are weakened. Constructs complete and coherent proofs using the definitions, ideas and theorems from the course. Applies ideas from the course to construct proofs that the student has not seen before.
• B: Applies the important theorems from the course. Constructs counterexamples when hypotheses are weakened. Constructs complete and coherent proofs using the definitions, ideas and theorems from the course.
• C: Applies the important theorems from the course when the application is direct. Constructs simple proofs using the de nitions when there are very few steps between the de nitions and the conclusions. Explains most important counterexamples.
• D: Can do some single step proofs and explain some counterexamples.
• F: Unable to do even single step proofs or correctly use de nitions.

Many courses combine pure and applied elements and the rubrics for those courses will have some combination of elements from the two rubrics above. Detailed interpretation of the rubrics depends on the content and level of the course and will be at the discretion of instructors. Whether to award grades of A+ is at the discretion of instructors.

Academic dishonesty