Multivariable Calculus I

Spring 2009 (1st half), Math 14a
Professor Francis Su
http://www.math.hmc.edu/~su/math14/

Course description

What is calculus? Up to now, you have mastered the theory of calculus of one variable. However, a quick glance around you reveals that you live in a world of three spatial dimensions (at least), and this world is replete with functions of more than one variable. This allows for the possibility of analyzing rates of change of three-dimensional objects in more than one direction. These three-dimensional objects include multivariable functions and vector fields.

So, in Math 14 and Math 61, you will learn the calculus of multivariable functions and vector fields. Topics in the Math 14 half-course include: multivariable functions and their derivatives, vector fields, gradient, divergence, curl, double and triple integrals, parametrized curves, flows, line integrals, Green's theorem, and flux integrals. (The topics in Math 61, the half-course that continues Math 14 in the sophomore year, will cover optimization, Taylor's theorem, Lagrange multipliers, Stokes' Theorem, and the Divergence Theorem.)

Office Hours

Prof. Su:
MON 10-11am, THU 1:30-2:30pm or by appt.
Olin 1269
 Academic Excellence Tutors:
SUN, MON, THU 8-10pm.
at Linde Activities Center, Riggs Room

Textbooks and course materials

There are two texts for the course. The primary book is Vector Calculus by Susan Jane Colley. A second book that will be used later in the course is Schey's Div, Grad, Curl and All That.

Homework

Homeworks will be announced on the course webpage: http://www.math.hmc.edu/~su/math14/. You will be told, as part of your homework, what sections to read to prepare for the lectures. READING AHEAD IS ESSENTIAL FOR SUCCESS IN THIS COURSE. My lectures will complement the reading by giving perspective.

Before tackling any problems, read the relevant section of the text and review your lecture notes. Feel free to consult me, the tutors, or your classmates about matters that are unclear to you.

Homeworks will be due on Tuesdays and Fridays at 2pm (in the bin outside my office door). NO LATE HOMEWORKS will be accepted except for medical or family emergencies (and must be approved in advance). Your lowest homework grade will be dropped.

You are encouraged to work cooperatively on your homework assignments with your classmates. However, you are expected to WRITE UP YOUR SOLUTIONS INDIVIDUALLY, i.e., you should understand your solutions well enough to write them up yourself. You should acknowledge any assistance you obtained from others or outside sources. The HMC honor code applies in all matters of conduct concerning this course. Note that copying work from published solutions (or the work of previous students) is a violation of the HMC Honor Code.

Exam Dates

Both exams will be take-home exams. Your first exam will be handed out Wednesday Feb 11, and due Friday Feb 13. Your second exam will be handed out Wednesday Mar 4, and due Monday Mar 9.

Evaluation of Learning

Traditionally, the purpose of a course grade is to evaluate your learning. Your course grade will be determined by your homework average and two exams, each contributing 1/3 towards you overall grade. However, there are two things that bear emphasizing.

One is that homeworks and exams serve a greater function than just evaluation; they are the vehicles by which you will learn. You learn through doing and exercises strengthen your understanding.

Second, I am proud to have you as a student regardless of how you perform in this class. You are more than a grade to me; you are individuals of great potential. I do hope to enhance your appreciation of mathematics' power and beauty and utility, and I only ask that you give me your best thoughts and efforts. I want you to let me know if there are unusual external pressures that are not contributing well to your learning.

Assignment Due Date

Assignments

Being able to direct your own learning is an important skill that we hope you will develop while at Harvey Mudd College. One component of being a self-directed learner is the ability to monitor your own understanding. You may wish to do more problems than those listed here and check your answers in the back of the book. Recommended problems are marked "R". They are not required and should not be turned in.

HW #1 due FRI 1/23 READ Colley 1.1-1.4 (mostly review)

DO

  • Problem A1. Using the reading as a guide, answer (in no more than 3 sentences): why is the interplay between algebraic and geometric descriptions of mathematical ideas useful? You might want to refer to an example (such as the book's example of judo and vector addition) to make your point.
  • Also 1.1 ( 11, 18, 21, R24, 25 )
  • 1.2 ( R3, R7, R13, 16, 24, 26 ),
  • 1.3 ( 8, 15, R21, R25 )

    Writing mathematics well is just as important as doing it, and graders may take points off for writing that they can't read or understand. You should always communicate your ideas in complete sentences. Please use these departmental guidelines when preparing your homework. Remember to indicate what section you are in (8am, 9am) next to your name.

    • HW #2 due TUE 1/27 READ Colley 1.5-1.6, 2.1-2.2. Also READ this handout on good writing.

    DO

  • 1.3( 25 )
  • 1.4 ( R5, R7, 17, 20, R21, 23, 25)
  • 1.5 ( R3, 9, 14, 24, R28 )

    Be sure to read the recommended problems and attempt them.
    Please use these departmental guidelines when preparing your homework. It is helpful to box your answers, but also to show all your reasoning. Do not forget to include helpful phrases such as "let" and "if" and "then" and "so" to connect your thoughts so the reader can follow them. Display important equations, underline or highlight important parts of your arguments, but write in complete sentences. (See this handout.)

    • HW #3 due FRI 1/30

    READ 2.2-2.3

    DO

  • 1.6 ( R10, R26 ), 1.8 ( Recommended: try True-False questions )
  • 1.9 ( 13, R15 )
  • 2.1 ( R2, 7, 14, 15, 18, R27, 29, 36 )
  • 2.2 ( R11, 14 )

    The last problem is there to encourage you to read ahead!

    • HW #4 due TUE 2/3 READ 2.4-2.5
    DO
  • 2.2 ( 23 )
  • 2.3 ( R1, R5, 6, 9, 18, R19, 23, 25, R26, R29, 30, R32, 34)
    • HW #5 due FRI 2/6 READ 2.6
    DO
  • 2.4 ( 2, R5, 14, 18, 21a )
  • 2.5 ( 2, R4, R5, 8, 10, R12, 16 )
    • HW #6 due TUE 2/10 READ 3.1, 3.3, 3.4
    DO
  • 2.5 ( R28, 34a[read text above problem 31], R35 )
  • 2.6 ( 5, R7, 11, 12, 13, 16, R20 )
  • 3.1 ( R1, 3, R4, R7, 10, 17, R19 )
  • 3.3 ( R2, R4, 9[describe in addition to sketch] )

    Begin preparing for your first midterm, handed out this Wednesday, and due back Friday. A good way to do this is to work all homework problems, study the relevant sections of the books, and study all your notes!

    • EXAM due FRI 2/13 REVIEW for your Exam:
  • READ your class notes and text!
  • SUMMARIZE important theorems and definitions from the NOTES and TEXT
  • MEMORIZE important formulas! (the exam is closed book, closed notes)
  • WORK problems from old homeworks (including RECOMMENDED problems)

    I would like you to meet with at least one other person in the class to study, and volunteer to EXPLAIN one concept to your study group. (You'll want to prepare in advance.) Here are some ideas:

  • what are the meaning and properties of the dot, cross products?
  • what's the difference between a graph and a level set diagram?
  • how do I parametrize a line? or a path?
  • what does a partial derivative mean? what is the chain rule?
  • how is the derivative matrix of f a "best" linear approximation to f?
  • how do I find tangent planes to graphs? tangent lines to paths?
  • what is the gradient of a function and what does it mean?
  • what is a vector field? what do the curl, divergence mean?

    There will be a question on the exam (worth 10 points) that will ask: Whom did you meet to study with? List all people. And what topic(s) did you explain to them? What topic(s) did they explain to you?

    • HW #7 due TUE 2/17 READ 5.1-5.3
    DO
  • 3.3 ( 18, 24, R25 )
  • 3.4 ( R3, 4, R7, 7, R8, 12, 13, R14, R16, 23 ) [Read problems 21-25.]
  • 5.1 ( R1, R11)
  • 5.2 ( 5, 10, R20 )
    • HW #8 due FRI 2/20 READ 5.4, 1.7, 5.5 (read Examples 10-17 only)
    DO
  • 5.2 ( 16, 23 )
  • 5.3 ( 1, R11, 13, 18, R19 )
  • 5.4 ( 4, 5, 8, R9, R11, 18, R23 )
    On problem 5.4.4, think about the meaning of the integral over this region
    • HW #9 due TUE 2/24 READ 5.5 (read Examples 10-17 only), 5.6, 6.1
    DO
  • 1.7 ( R1-R17[do enough for practice], 24, R27, 28, 33, R35 ), 1.9 ( 38 )
  • 5.4 ( R19, R23, 25ab )
  • 5.5 ( R15, R20, 23, R24, 25, 26, R27, 28 )

    The diagram of Problem 5.5.23 has an error: the line y=sqrt(3x) should say y=sqrt(3)x.

    • HW #10 due FRI 2/27 READ Colley 6.1-6.2 and Schey's "Div, Grad, Curl and all that", pp.1-8, Chapter III.
    Schey will be used in the final week of this class and throughout Math 61. There won't be many problems from Schey but it is good for a different perspective: it has more of a physical approach with slightly different notations: (t-hat ds) is the same as (ds-vector).
    DO
  • Colley 5.6 ( 3, 8, 12 )
  • Colley 6.1 ( 1, 2, 7, 8, R11, R13, 15, 24 )
  • Schey III ( R4, R7 )
    • HW #11 due TUE 3/3 READ 6.3 and Schey pp. 63-81.
  • Colley 5.4 ( 20 )
  • Colley 6.1 ( R11, 21, R25, R29, 30 )
  • Colley 6.2 ( R1, 4, R8, 11, 13 )
  • Colley 6.3 ( 1 )
  • and Problem A: Prove Theorem 4.3 on page 218 (try to do it without looking at the proof), then prove Theorem 4.4 on page 218.
    • HW #11 due TUE 3/3 READ 6.3 and Schey pp. 63-81.
  • Colley 5.4 ( 20 )
  • Colley 6.1 ( R11, 21, R25, R29, 30 )
  • Colley 6.2 ( R1, 4, R8, 11, 13 )
  • Colley 6.3 ( 1 )
  • and Problem A: Prove Theorem 4.3 on page 218 (try to do it without looking at the proof), then prove Theorem 4.4 on page 218.
    • HW #12 due FRI 3/6 READ 6.3 and Schey pp.86-91, 115-121.
  • DO Colley 3.5 ( R15, R17), 3.6 ( R1, R3 )
  • DO Colley 5.7[explain why] ( 7, 25 ), 5.8 ( 1, R2, 7 ),
  • DO Colley 6.3 ( 3, 4, R7, 17, R18, 19 ) 6.4[explain why] ( 3, R5, 7 ) 6.5 ( R1 )

    The problems above will help you review for the final exam. Study class notes, texts, and work problems--- odd-numbered exercises and true/false questions at the end of each chapter of Colley are good places to start. The exam will focus on material since the last exam, but any concept from the course may be tested.