Comments (Current)

• 2/19: This week, we covered orientations, and calculating surface integrals with coordinate elements (p. 1119-20). The textbook failed to address the orientation issue adquately. Please see my note on orientation. I will update the preparating midterm2 this weekend.
• 2/12: Various aspects of applications of Green's theorem discussed. Flux also introduced. div and curl introduced.
• 2/10:The statement of Green's theorem; and a proof in a simple case. The other two emphasizes are on: 1. induced orientation of the boundary of a domain; 2. The understanding of Greens' theorem using differential forms. For this week, one needs to be able to calculate the differential of any one form. A note on differential form can be found in coursework.

Tentative schedule, subject to change

Week One	Rieview: Riemann Sums 13.1 Double Integrals 13.2 Double intgrals over general regions
Week two	13.3 Area and Volume by Double Integration. 13.4 Double Integrals in Polar Coordinates.
Week Three	13.5 Applications of Double Integrals. 13.6 Triple Integrals.	1/18, Holiday 1/24,2/26, drop deadline
Week Four	11.8 Cylindrical and Spherical Coordinates. 13.7 Integration in Cylindrical and Spherical Coordinates. 13.9 Change of Variables in Multiple Integrals.	1/28, Midterm 1, 7-8:45PM
Week Five	14.1 Vector Fields, Gradient, Divergence, Curl and Del operators. 14.2 Line Integrals 14.3 The Fundamental Theorem and Independence of Path.
Week Six	14.3 The Fundamental Theorem and Independence of Path. 14.4 Green's Theorem Topic A: Differential forms on xy-plane.
Week Seven	13.8 Surface Area. 14.5 Surface Integrals	2/15, Holiday
Week Eight	Review 14.6 The Divergence Theorem	2/28, Withdraw deadline 2/25, Midterm 2, 7-8:45PM
Week Nine	14.7 Stokes' Theorem
Week Ten	Differential forms. Review and applications.
	Final Exam Monday March 14th 7-10PM

 

Comments (after lectures)

• 2/8: Independent of path; conservative fields, and curl free fields are covered. Simply connected region is introduced. For simply connected region, see <Simply connected definition>. On Wed and Friday, we will cover Green's theorem, etc., and will introduce differential forms on 2-plane.
• 2/5: This week, we covered line integrals. The mass does not depend on the orientation of the path, but the work, and the integral with respect to the coordinate variable does. We will continue with the notion of independence of path, conservative fields next week.
• 1/29: We covered the topic of change of coordinates in double and triple integrals. Note that we take the absolution value of the jacobian because we are dealling with area of volume element, no orientation for the moment. (We will postpone the surfacr area until whem we do surface integrals. See the change to the Syllabus.)
• 1/27: Spherical coordinates introduced, and explained.
• 1/25: Cylindrical coordinates introduced. It is merely dz plus polar coordinates.
• 1/22: We covered Section 13.5. The two topics covered are how the formular for centroid is derived, and how to prove the first Theorem of Poppus. (The fomular in this section one needs to "memorize" are the centroid and first theorem of Poppus.
• On Midterm: See the course website.
• 1/20: We covered triple integration (Section 13.6). Key: try to project the region to a coordinate plane, to get a region, and determine the upper and lower bound of the integrating lines intersecting the region. There are three choices of coordinate planes to project to, and the resulting integral bounds can have various level of difficulties. Again, draw picture, put information on the picture, and be patient.
• 1/15: We covered more extensively the double integral in polar coordinates. The two in gredients are: 1. the area element in (r,\theta) is via rdrd\theta. On Wed. next week, we will cover triple integral, and application of double integral on Friday.1/13: The focus is on setting up the couble integral to evaluate volume of solids. This is the prelude of triple integration. The geometric intiution will be very crucial for setting up theintegral. We began integration with polar coordinate. We will contiue on Friday. Review Section 9.2, polar coordinates.
• 1/11: The focus is to set up iterated integrals by splitting the regions into simple ones, and how to read the region knowing the interated integral limits. This Wed., we will work on volume, and integration via polar coordinates. This time, visualizing the intersection of a quadric surface with a plane or with another quadric surface is crucial to set up the integral for volume. Review Section 11.7, quadric surfaces. Also, review Section 9.2, polar coordinates.
• 1/8: The focus of today's lecture is on how to transform a double integral to an interated integral. Please read Theorem 1 on page 1008 again, and connect formula (4) and (5) with what I said in the class. Always draw picture,draw vertical (when integrad y first) or horizontal (when integrad x first) arrow to indicating which variable to integrate first, and from where to where. We will contiue with double integral in xy-coordinate next lecture.