Zvezdelina Entcheva Stankova

Visiting Professor


713 Evans Hall

Department of Mathematics

University of California at Berkeley, CA 94720-3840

Tel: 510-642-3768, Email: "last name, lowercase" at "university name" dot edu

Office Hours: TTh 9:30-10:30am (first in front of 155 Dwinnelle right after lecture, then Evans 713), T 3:30-4:30pm in Evans 713.

Webpage: http://math.berkeley.edu/~stankova

Course Syllabus for MATH 53, Spring 2016

(updated 1/22/2015)

GSIs finalized office hours will be revised here. Check them out and keep them with you at all times. You are welcome to visit any GSI's office hours, along with the professor's office hours.

Christopher Eur's revised office hours (as of 1/24/2016): M 5-6pm, T 12:30-1:30pm, W 11am-12pm.

Kai-Chieh Chen's revised office hours (as of 1/25/2016): M 2:30-4pm, W 3-4:30pm.

All other GSI's office hours are as in the syllabus.

For enrollment questions:

Do NOT contact the instructor or the GSIs. We have no control over enrollment. Contact instead Thomas Brown, 965 Evans, 10:30am-12noon or 1-5 pm. You need to see him in person (not by email!) to resolve enrollment questions. Students who wish to switch sections with open spaces can use the "Switch Section" link in TeleBears. Thomas Brown cannot move students into sections that are full. However, he can often swap students between sections. If two students are willing to swap sections, both students need to go to Thomas and he will try to switch them.

Email is ONLY for emergencies (e. g., medical and family emergencies).

Email is NOT for resolving enrollment questions, or asking for letters, or for discussion of midterm results, or for any discussion about how the student is doing in the class or how to improve. For such questions, students need to come see me in person in office hours (bringing all necessary documentation with you). You are also welcome to visit any GSI's office hours to discuss math questions or how to improve in the course; the GSIs are very qualified to discuss any math question. This is clearly written in the syllabus and is discussed in detail during the first lecture.

Homework Assignments, MATH 53

HW1A. Read: 10.1. Curves defined by parametric equations. 10.5. Conic sections (concentrate on recognizing the type of conic section: parabola, ellipse, or hyperbola, and sketching graphs)

Write Exercises:

10.1. #6,8,10,12,14,16,20,22,24,38. Bonus: #31,46(a)(c).

10.5. Just sketch the graphs: #6,14,22. Bonus: #8,16,24 (complete the square(s) in the bonus problems).

HW1B: Read: 10.2. Calculus with Parametric Curves: Tangents (in polar coordinates), Examples 1 and 2, pp. 650-651. 10.3. Polar Coordinates

Write Exercises:

10.2. #2,6,8,14,18 Bonus: #28,30

10.3. #2,4,6,10,16,18,24,30,56,60 Bonus: #14,28

Notes from class 1/21/16 and office hours:

1. The "mystery" of why the "polar" angle theta for point A(7 cos(45), 5 sin(45)) on our ellipse did not match the "parameter" angle t? There is no mistake in what we did in class. Indeed, the polar angle theta for A is arctan(y/x)=arctan(5/7). This angle theta is different from t = 45 degrees = pi/4 for the simple reason that the parameter t is different from the angle theta. When we parametrized the ellipse x^2/49 + y^2/25 = 1, we used the original parameter t from the unit circle. However, when the unit circle is rescaled to the ellipse, i.e., (u,v) --> (x=7u, y=5v), the angle with the x-axis changes. In our example, take point B(cos(45),sin(45)) on the unit circle: the angle (whether "polar" or "parameter") is 45 degrees; yet, when we rescale to get from point B to point A, the angle with the x-axis changes to theta = arctan(5/7), which is the polar angle and not the original 45-degree angle used as a parameter.

2. Regarding the polar equation of the line y = (square root of 3) x. If we want one polar equation, then we have to allow r to be negative in r = pi/3. If we want to keep r non-negative, we will need two polar equations: r = pi/3 and r = 4 pi/3; each of these last two equations (with r non-negative) describes a ray, and together they describe the whole line.

3. Review and be ready to recognize and apply some standard trig. formulas. For example, today in class we used the formulas for double-angles:

sin (2x) = 2 (sin x)(cos x)

cos (2x) = (cos x)^2-(sin x)^2 = 1- 2 (sin x)^2 = 2 (cos x)^2 -1

4. Regarding using LH to verify that the cycloid has vertical tangents at theta = 0, 2 pi, 4 pi, etc.: the formula for the tangent slope is sin(theta)/(1-cos(theta)). When theta = 2 pi, for example, the ratio becomes 0/0. Thus, we take the limit of sin(theta)/(1-cos(theta)) as theta approaches 2 pi (from the left and separately from the right), apply LH to replace top and bottom by their derivatives: cos(theta)/sin(theta) (we still have the limit at this stage), and finally, we plug in theta = 2 pi to get 1/0 (0^+ if approaching 2 pi from the right, and 0^- if approaching 2 pi from the left). In any case, we get the tangent slope to be plus or minus infinity, which indicates a vertical tangent. Try to repeat the calculation from class that did NOT involve LH, but simplified first the tangent slope and then showed directly that where the tangent slope was infinity (or undefined).

5. The example r = 4 sin (theta): it is a single circle x^2 + (y-2)^2 = 4, as we concluded in class. It is not two circles on top of each other, although it is good to confirm this. In class we saw what happens when theta was between 0 and pi. When theta becomes larger than pi, e.g., theta = 5 pi/4, then r is negative: r = - 2 (square root(2)). When we make r positive, we need to subtract pi from theta, so we land on the point (r = 2 (square root(2)), theta = pi/4), which is one of the points on our circle that we already saw earlier. Thus, when theta becomes larger than pi, we start going over the same circle a second time, etc.

HW2A. Read: 10.2. Calculus with Parametric Curves (Review and finish). 10.4. Areas and Lengths in Polar Coordinates

Write Exercises:

10.2. #32,36,38,42,48,58 (in #38,58: just set up the integral), #62,66. Bonus: #73,74.

10.4. #4,6,8,12,18,28,32,38,46,48. Bonus: #36,44.

General notes on bonus HW problems: Bonus HW problems will be included in the weekly HW solutions. Ordinarily, they won't be included in quiz for that week, but they may be included in exams (or even later quizzes, if suitable), so you need to study their solutions carefully. Keep in mind that a "bonus" problem earlier in the semester may not look as hard later on in the semester.

HW2B. Read: 12.2.Vectors

Write Exercises:

12.2. #2,4,6,8,14,16,20,22,24,26,28,30,32,42,44,51. Bonus: #40; Prove that the three medians in a triangle intersect in one point (called the centroid of the triangle) and that this point divides each median in ratio 2:1 counted from the vertices of the triangle. (A median connects a vertex of a triangle with the midpoint of the opposite side.)

Note: Problem 10.2. #36(c) was assigned unintentionally. We will not be testing on exams about centroids of general figures. The only centroid for now that you should be aware of and be able to solve problems with is the centroid of a triangle (as we did in class). The first video below ("Triangle has a Magic Highway") and its extra footage will tell you more on that topic.

HW3A. Read: 12.3. The Dot Product (including Ex.7 on work)

Write Exercises:

12.3. #6-14,18,22-30,36,38,42,50,56,58*. Bonus: #61,62.

HW3B. Read: 12.4. The Cross Product. 12.5. Equations of Lines and Planes (only the first section on Lines, including Example 3 on p. 826)

Write Exercises:

12.4. #2-20,30,36,38,44. Bonus: #47,48.

12.5. #2,4,10,14,18,20,22. Bonus: #12.

Solutions to HW3. The HW solutions will be taken off the web in a week and will not be distributed in any form later on, so make sure you download and study the HW within the next few days.

HW4A. Read: 12.5. Equations of Lines and Planes (finish, concentrating on Planes and Distances, pp. 827-830). 12.6. Cylinders and Quadric Surfaces

Write Exercises:

12.5. #26,28,38,40,46,56,66,70,72,74,76 (in #76: use formula from #75). Bonus: #62,75.

12.6. #2,6,10,18,34,36,44,46,48. Bonus: #52.

HW4B. Read: 13.1. Vector Functions and Space Curves. 13.2. Derivatives and Integrals of Vector Functions

Write Exercises:

13.1. #2-12,16,18,28-32,42,44. Bonus: #46,48,50.

13.2. #2-10,18-22,26,28,34,36,40,42. Bonus: #16 (in two ways!),55.

General note on HW problems: Ordinarily, only even-numbered problems are assigned, unless an odd-numbered problem is explicitly included in HW. Thus, for example, #2-12 means #2,4,6,8,10.

Videos on Game and Geometry Puzzles from the Numberphile Channel with Zvezdelina Stankova, directed by Brady Haran, containing some proof and problem-solving ideas that come handly in a variety of math areas:

Triangle has a Magic Highway (Euler Line): see how a triangle's center of mass relates to other centers of the triangle

Three Squares

Freedom for the Clones

Berkeley Math Circle