CHAPTER 1: Vectors.

• Homework 1: Due Monday Jan 29, 2007, 11:00 am ( Some (fabulous!) solutions)
  • 1.2. 6, 7, 8
  • 1.3. 3, 7
  • Prove formula (16) in the notes using definitions (11), (14)
  • 1.4. 4, 5
  • 1.5. 2, 5

You should be able to log into the Math 321 webwork site using your 10 digit student ID (no dash, no space) as login name and password, unless you are not registered or have already changed your name and password as you can do once you're in Webwork.

When in WeBWork, don't click away frantically using trial-and-error till you are lucky enough to hit the correct solution, have a sheet of paper and a pencil handy in addition to the lecture notes and your class notes. Make an effort to visualize the vector problems.

• WeBWork 1: Due Fri Feb 2, 10pm (see in WeBWork)
• WeBWork 2: Due Sun Feb 4, (see in WeBWork)
• WeBWork 3: Due Wed Feb 7, (see in WeBWork)
• WeBWork 4: Due Sun Feb 11, (see in WeBWork)

WARNING: WeBWork is used to give you drills, automatic feedback and automatic collecting and grading. It is NOT a substitute for exercises in the notes. EXAMS will be more similar to exercises in the notes as well as to reasoning and manipulations used to derive results in the notes themselves (not just exercises). So your first priority is to study and understand the notes in depth. WeBWork is just one more tool to help you do that.

• Exam type problems for 1.6, 1.7: (NOT due but strongly suggested, did I say `exam type'?)
  • 1.6. 1-4 (1,2 & 4 were solved in class, if you understood 4, then 3 should be a piece of cake. The notes from formula (39) to (44) are a great series of solved exercises! Read the notes slowly, try to understand where we start and where we want to go. Try to deduce the next step on your own before reading it in the notes. Understand, deduce, verify, go on.)
  • Derive & Digest formula (54), (55) and (56) in the notes. Learn or remind yourself of the good old 3 by 3 algebraic determinants and their fundamental properties (57), (58), (59) which are the direct equivalent of (49), (50) and (51).
  • 1.7. 1-13, [14, 15, 16 are more advanced but look at the pretty parallel between (16) and (10). This goes on to higher dimension and it's a hint about how you measure the `areas' of hyperparallelograms in n-space]

• Section 1.8 (lines, planes, etc): All the examples/problems in section 1.8 are fundamental. You cannot do much if you're not comfortable with equations of lines, planes, spheres, etc. these are basic geometric objects. Finding the distance from a point to a line or to a plane are also basic geometric operations.

• Section 1.9, Vector functions of a scalar variable i.e. curves. The 3 exercises on the bottom of page 23 are important, they come back as `easy' steps of many more complex problems i.e. don't digest them now, pay later.

• Section 1.10, Motion of a particle. You have probably seen this in dynamics (EMA 202) or Mechanics (PHYS 311). Make an effort to digest rotation about an arbitrary axis. This is where we find out whether you really understood the cross product. Look back at section 1.5, rethink parallel and perpendicular components. The notes need to be illustrated by the plots made in class, hopefully you wrote them down or can make your own. There are additional notes on that topic, however those notes were written for the slightly more complex problem of motion of a charged particle in a magnetic field but that is a closely related problem and the beautiful color pictures in there should be helpful. (You're welcome. Yes it does take a lot of work and effort to make all those.) Those notes complement the lectures. There is a lot of overlap with the lectures but the lectures and notes are not identical. Kepler-Newton problem is also fundamental, beautiful and historically important.

• Sections 1.11 and 1.12, good stuff but we cannot do everything. SKIP in Math 321, hopefully this is studied in Mechanics (Physics 311) and Advanced dynamics (EMA 542).
• Section 1.13: cartesian coords. Basic, need to understand how to go from vector notation to cartesian and back.

CHAPTER 2: Matrices

WARNING: The notes do not yet have all the helpful sketches and figures made in class, so you need to fill that in from your own lecture notes.
[The notes for Chapter 2 do not use the summation convention, so all sums are written out explicitly to help you out since we're introducing yet another notation: the linear algebra notation].

• You need to understand the derivations in section 2.1. Including being able to do them using the summation convention (as done in class).
• You need to understand the linear algebra notation and its connections with the indicial notation, need to know how to multiply matrix by vector, matrix by matrix, what is a `transpose', what is a symmetric matrix, an anti-symmetric (or skew-symmetric) matrix, an orthogonal matrix (section 2.2).
• Need to understand elementary rotation matrices (that is a rotation of basis about e₁ or e₂ or e₃)
• Need to understand that any 3-by-3 proper orthogonal matrix is a general rotation and that it can be represented in terms of a product of 3 elementary rotation matrices. The angles associated with these elementary rotations are called Euler angles. How does one show/derive that?
• Exercises: try all.
  2.2.9, and 10, 11, 12, allude to the linear algebra way to rotate a vector about a rotation axis. It involves doing
  1. TWO basis rotations (and the corresponding component transformation) using matrices Q₃ and Q₂, rotate about original e₃ first, that's Q₃, then rotate about new e₂ to align new e₃ with rotation axis, that's Q₂
  2. now that we have the components in an appropriate basis, the actual vector rotation is easier to do (rotate about new direction 3 by given angle), that's another rotation matrix, R say, (but watch out! this one is to actually rotate the vector not the basis )
  3. transform back to original coordinates. That involves multiplying by the transposes for Q₂ and Q₃.
  So the entire rotation consists of multiplying the original vector components by [ Q₃^T Q₂^T R Q₂ Q₃]. Five rotation matrices are involved! The basis-independent approach we used earlier (using parallel and perp components) is more elegant and simpler, if you can think in terms of vectors v as geometric objects and can use your cross-product. If you're stuck with cartesian components and see vectors only as list of components (v₁, v₂, v₃ ), then your need rotation matrices, 5 of them!
• 2.4.1 is something we already saw a couple of times, so the third should be a charm!

CHAPTER 3: Vector Calculus

• Curves and curve integrals (Sect. 1.1, 1.2) You must know how to deduce/construct the general parametric vector equations of planes, circles and ellipses. You must understand formula (1)-(7) in section 1.2. Several of the exercises were actually solved in class. Try out
  • 1.2.1, 2, 4, 5, 6 Think about #3 after solving #2. You want to understand how you would go from intrinsic coordinates (x₁, x₂) to (x,y,z) and vice versa.
• Surfaces and surface integrals (sect. 1.3, 1.4) You must know how to deduce/construct the parametric vector equations of planes, spheres and ellipsoids. You must understand the meaning of surface coordinates, coordinate curves, normal vectors, surface elements. Again some of the exercises were done in class.
  • 1.3.1, 2, 3, 5 think about 4 after solving 3, it's a good exercise but long since it has several parts, you want to understand how you would go from intrinsic coordinates (x₁, x₂, x₃) to (x,y,z) and vice versa . #2 is a `big' problem with many parts that you must understand in depth since it is about those important spherical coordinates.
  • 1.4.1, 2, 3, 4, 5
• Volumes (sect 1.5) Must understand coordinates curves, coordinate surfaces, general line element, surface elements, volume element. Spherical coordinates.
• Curvilinear Coordinates (sect 1.6) Parametrization of curves, surfaces and volumes are examples of `mappings' and `curvilinear coordinates'. The parameters describing those 1D, 2D or 3D objects are coordinates for points on these objects. The main new thing here is that we switch to a more abstract but more manageable indicial notation and we also discuss the special case of orthogonal coordinates, in that case the metric factors h₁, h₂, h₃ determine (almost) everything we need to know to do calculations in those coordinates.

• Homework 2: Due Monday Mar 12, 2007, in discussion or class whichever comes first (note: not all problems will be graded)
  • 1.2. 5, 6 Mon 3/12/2007: interpret geometrically before jumping into calculations
  • 1.3. 2, 5
  • 1.4. 2, 4, 5 Mon 3/12/2007: 4 and 5 solved in class, digest them well
  • 1.5. 1 (but for coords (15) not (14)), 3
  • 1.6. 2

• Change of variables, Jacobian matrix and Jacobian determinant (sect 1.7) Again closely related to parametrizations of surfaces and volumes and to `curvilinear coords'. But more general, for instance in thermo: switch from pressure and volume variables p, V to temperature and entropy T, S , especially for the Carnot cycle.
  • 1.7. 1, 2, 4, 5, 6

• Gradient (sect 2.1) Must understand geometric meaning and be able to use it to directly calculate some gradients such as examples at bottom of page 15
• Homework 3: Due Monday Mar 19, 2007, in discussion or class whichever comes first (note: not all problems will be graded)
  • 1.6.3, 4 (1.6.3 should read w = z ).
  • 1.7.1, 4, 5
  • 2.4.6, 7, 8, 9 [Note: 7, 8 and 9 were basically solved in class on Mon Mar 19, 2007, 9 was redone in class on Fri Mar 23 in conjunction with problem 3.7.3]

• Fundamental Theorem of Calculus Must understand how the familiar FTC for one variable leads to (80) which can be written as (81), likewise (82) which can be written as (83). [Exam question could be to ask you to prove (81) or (83) or (84) or (85). Class voted to keep proof of Stokes theorem on page 24 as a potential exam problem ;)]. Exercises: we showed how (84) could be used to calculate areas of complicated domains defined by their boundary. We did 3.7.1 (i) in class. Wed Mar 21: exercises 3.7.1-4 in addition to understanding the notes so you could prove (81) or (83) etc.
• Homework NOT due:
  • 3.2 You've been computing easy integrals using a well-known recipe since your high school days. Do you understand where that recipe comes from?
  • 3.3. Can you prove (81), (83) or the generalized "s" version we saw in class?
  • 3.4. Do you understand what (84) means? Do you know it? Can you prove it? Can you use it? You must know and understand (86): Stokes' theorem.
  • 3.5. Done slightly differently in class (used "s" there, may be a better way, but here's another). Can you show (103), (104), (105)? Do you understand their meaning?
  • 3.6 Must understand (107), (110) and (111). Then (113)-(118)
  • 3.7 Must attempt/study/digest exercises 1-9 at least. 10 and 11 not too hard. 12 very, very interesting but not trivial, that's A+ material. Exercise 1 was partly done in class on Wed Mar 21. #3 is a very important problem done in class on Fri Mar 23. It is similar to #9, but #3 is Stokes, #9 is Gauss, don't mix up the two. #4 n should be omega . Several of these exercises will be done in class on Mon Mar 26 and Wed Mar 28.