Vector-Calc-S17

Vector Calculus Syllabus Spring 2017

MAT226 Vector Calculus: 4 hours, 4 credits. Vectors in two and three dimensions, equations of lines and planes, functions of several variables, partial differentiation, directional derivatives, gradients, optimization with Lagrange multipliers, multiple integration, line integrals and vector fields

Prerequisite: A grade of C (or better) in MAT 176.

Professor Sormani sormanic@gmail.com

Office Hours M/W 5:30-6:00 pm & 7:45-8:15 pm in 200A Gillet Hall

Grading Policy

Expectations: Students are expected to learn both the mathematics covered in class and the mathematics in the textbook and other assigned reading. Completing homework is part of the learning experience. Students should review topics from prior courses as needed using old notes and books.

Homework: Approximately two hours of homework will be assigned in each lesson as well as additional review assignments over weekends. You are responsible for checking your answers and doing the homework. If you do not keep up on your homework you will fail the course. Some of the homework is less specific suggesting that you do problems of certain types that you find at the end of the chapter. This is because different editions of the textbook number the questions differently.

Exams: There will be four exams and a final exam.

Grades: Each Exam is worth 15% and the Final is worth 40% If an exam is missed or you perform poorly on an exam it may be taken a second time with two weeks but the new grade will be multiplied by 0.8.

Materials, Resources and Accommodating Disabilities

Textbook: Larson, Hostetler and Edwards, Calculus: Early Transcendentals Ed. 4, Houghton Mifflin OR Larson, Hostetler and Edwards, Calculus: Early Transcendentals Special Edition of Lehman College 175-176 Ed. 5, Houghton Mifflin

Technology: Students should purchase a basic scientific calculator able to compute trigonometric and exponenetial functions, but unable to complete algebraic manipulations and take derivatives.

Tutoring: Departmental tutoring is available in the Math Lab on the 2nd floor of Gillet. Reliable Web Resources: See http://comet.lehman.cuny.edu/calculus

Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441.

Course Objectives:

At the end of the course students should be able to:

1. Graph and determine the equations for lines and planes (as part of dept objectives a & b)

2. Compute sums, differences, dot products and cross products of vectors (a)

3. Determine velocities and accelerations of vector-valued position functions (a, b & c)

4. Find level sets, gradients and tangent planes to functions of several variables (a, b & e)

5. Apply the method of Lagrange Multipliers (a,b & c)

6. Apply Fubini's Theorem and Green's Theorem to integrate functions and fields (a, b & e)

These objectives will be assessed on the final exam along with other important techniques.

Course Calendar

Lesson I (Mon 1/30): Vectors, Plotting in 3D, 11.1­-11.2

All odd problems in 11.1­-11.2

Lesson II (Wed 2/1): Dot and Cross Products 11.3­-11.4

All odd problems in 11.3­-11.4

Review Differentiation 3.1­-3.2

Lesson III (Mon 2/6): Parametric Equations and Polar Coordinates 10.2, 10.4

10.2/ 1­-50 odd, 10.4/ 1­-50 odd (do over the weekend)

Lesson IV (Wed 2/8): Lines and Planes 11.5

11.5/ do five problems on parametric equations for lines, do five on equations of planes

Lesson V (Wed 2/15): Hyperboloids, Paraboloids 11.6, Cylindrical and Spherical Coordinates 11.7

11.7/ 1­-71 odd, sketch these, 11.6/ 1­6, 9,11,13,15

Lesson VI (Wed 2/22): Exam I on 11.1-11.7, 10.2, 10.4,

Review Limits and Continuity 2.3­-2.4 and Differentiation 3.3-3.4

Lesson VII (Mon 2/27): Vector valued functions, limits and continuity 12.1

12.1/ 1­-13 odd, sketch 3 curves using the function, evaluate 3 limits, find 2 intervals of continuity,

Review Integration 5.1, 5.5

Lesson VIII (Wed 3/1): Differentiation and Integration 12.2

12.2/ 1­-17 odd, find 3 indefinite integrals, find two definite integrals

Lesson IX (Mon 3/6): Velocity and Acceleration, 12.3

12.3/ 1­-16 odd, do 4 projectile motion, 1 cycloidal motion and 2 circular motion problems

Lesson X (Wed 3/8): Tangent Vectors and Arc length 12.4­-12.5

12.4/ 1­-16 odd, 19, 31, 33, 35, 45; 12.5/ 1, 3, 5

Doing these problems reviews many of the topics on Exam I.

Lesson XI (Mon 3/13): Exam II on 12.1-12.5

Lesson XII (Wed 3/15): Functions of several variables 13.1

13.1/ 3 find and simplify function values, 3 describe domain and range,

13.1/ 2 do problems on contour maps, descriptions of level sets, sketching graph of levels, applications

This is all good for practicing for retaking Exam I but also practice old work as well.

Lesson XIII (Mon 3/20): Partial derivatives 13.3, (13.2 if time)

13.3/ 9­-25 odd, 37, 53, 65, prepare to do retake of Exam I if you scored below 80%.

Lesson XIV (Wed 3/22): Exam I Retake and in class Extra Credit for those who don't need the retake

13.3/ show mixed derivatives are equal, do problems on Laplace’s equation, wave equation, heat equation, marginal productivity, ideal gas

Lesson XV (Mon 3/27): Chain Rule 13.5 and Gradients 13.6

13.5/ 1­-11 odd, 23, 27, 31; 13.6/ 1, 3, 13, 15, 21, Review HW: Extrema in 4.1

Lesson XVI (Wed 3/29): Tangent Planes 13.7

13.6/23, 25, 27, 31, do problems on normal to level, topography, heat seeking, meteorology

13.7/ 5, 7, 9, 17, 19, 21

Lesson XVII (Mon 4/3): Quick Exam III on 13.3, 13.6, and 13.7, and Lecture on Extrema and Saddle Points 13.8

13.8/ 1, 3, 25, 27, (ed4: 41, 45, 53, 57) or (ed5: 55, 37. 43, 45, 47)

Lesson XVIII (Wed 4/5): Optimization 13.9

13.9/ Max volume package, max volume ellipsoid, max revenue, max profit, min cost

Photos of the semester's blackboards are here.

Lesson XIX (Wed 4/19): Lagrange Multipliers 13.10

13.10/ 1, 3, 5, 7,

Lesson XX (Th 4/20): More Lagrange Multipliers 13.10

13.10/ do max vol, min cost, refraction of light, production level, putnam challenge

Lesson XXI (Mon 4/24): Exam IV on 13.8. 13.9, 13.10 and Retake Exam III

Lesson XXII (Wed 4/26): Calculus II Review Lesson

Review HW: Definition of Integration 5.2­-5.3

Review HW: Techniques of Integration 5.5

Lesson XXIII (Mon 5/1): Iterated Integrals and Area 14.1 Double Integrals 14.2

14.1/ 1­9, 11­-15, do 2 areas of region, 14.2/ 1, 3, 7, 9, 13, 15,

Lesson XXIV (Wed 5/3): Integration and Polar Coordinates 14.3

14.2/ 4 do problems on volumes of sketched regions, 3 set up and evaluate double integral, Putnam challenge,

14.3/ 1, 3, 5, 7, 9, 11, 3 use double integral to find the area of the shaded region problems

Review HW: Polar Coordinates 10.4, study to retake Exam IV

Lesson XXV (Mon 5/8): Retake Exam IV and in class Integration Extra Credit on 14.1-14.3

Students who have taken linear algebra already should read 14.8

Review HW: Vector­ valued functions 12.2-­12.3

Lesson XXVI (Wed 5/10): Vector Fields and Line Integrals 15.1­-15.2

15.1/ 1, 3, 5, 7, 9, 11, find gradient vector field, verify conservative and find potential, find curl, find divergence, 15.2/ 1, 3, 5, 7, 9, 27, 35, 39, and study to retake Exam II.

Lesson XXVII (Mon 5/15): Path Independence and Green’s Theorem 15.3­-15.4 and Retake Exam II

15.3/ 1, 3, 5, 7, 11, 15b, 19a, 25, 27, 35,53, 15.4/ 1, 3, 5, 7, 11, 21,

Lesson XXVIII (Wed 5/17) : A Brief Survey of Surface Integrals and Review for final

14.5, 15.5, 15.6, 15.7, 15.8 are highly recommended

Extra Meetings (Friday 5/19): 3-4 pm and 6-7 pm in the usual classroom on Integration

Photos of the entire course by Jake are here.

Format of the Final:

Part I: on Exams I-IV

10% Graph a curve (+1), find the tangent vector (+3) and the tangent line at a pt (+5) and graph (+1) as in Exam II

10% find position from acceleration as in Exam II

10% Graph a surface (+1), find the normal (+3) and tan plane at a point (+5) and graph tan plane (+1) as in Exam III

10% find critical points of f(x,y) by solving grad f = 0 (+5) and check which is a saddle or rel max/min using the second derivative test (+5) as in Exam IV

10% find the maximum of a function subject to a constraint equation, set up Lagrange (+5) Solve (+3) Finish (+2) as in Exam IV

Part II On Integration (14.1-14.3)

10% find the average of a function f(x,y) over a rectangular region (14.1-14.2)

10% find the area of a region by carefully choosing the bounds of integration using rectilinear coordinates dx dy (14.2)

15% find the area of a region using polar coordinates to complete the integration (14.3)

15% find the volume under a surface using any method you wish (14.2-14.3)

Extra Credit on 15.1-15.3: div, curl, line integrals, potentials (similar to the div curl EC exam)

Extra Credit on Surface Integrals 14.5, 15.5-15.8 (those who wish for letters of recommendation may do this at a later date)

Final Exam: The Final Exam will be given during Finals Week Monday May 22 6:15-8:15 pm covering the entire course especially topics needed in future courses. Half of the final will be on 14.1-3, Extra Credit of the Final will be on 15.1-4. See description above.

This syllabus and others are available at: http://comet.lehman.cuny.edu/calculus/. Department of Mathematics and Computer Science, Lehman College, City University of New York