CHAPTER 1: Vectors.

• Homework 1, Tue 9/4/07 : READ SECTIONS 1.1, 1.2( Some (fabulous!) solutions)
  • 1.1.4
  • 1.2. 1, 3, 5--8
• Homework 2, Thur 9/6/07 : READ SECTIONS 1.3, 1.4. You must know dot product in Rⁿ, you *can* skip discussion of norms on pages 9, 10. Make sure you digest formula (14) and (15) they come up over and over and over.
  • 1.3. 1--8, do 1.3.4 two ways: geometrically and vector-algebraically
  • Prove that definition (17) satisfies the 5 dot product properties. What is the angle between (1,4,3,2) and (4,3,2,5) in R⁴?
• Homework 3, Tue 9/11/07 : READ SECTION 1.5. Make sure you have a good geometric understanding of cross product.
  • Explain geometrically why a x ( b + c ) = a x b + a x c (i.e. make a sketch that explains this from geometric definition of cross product).
  • When discussing ( a x b) x a in class, we got |(a x b) x a| = |a|^2 |b_perp| (note the bars |.| indicating `length of' or `magnitude of', then I dropped some of the bars and wrote (a x b) x a = |a|^2 b_perp. Why is that correct? If |u| = |v| does this mean that u = v? Or if |u| = |v| |w|, does that mean that u = |v| w? Explain.
  • 1.5. 1--6
• Homework 4, Tue 9/18/07 : READ SECTION 1.6. Index notation, Einstein summation convention
  • From formula (25) in the notes (double cross product), which we proved geometrically, deduce a simpler expression for &epsilon_ijk &epsilon_ilm
  • 1.6. 1--4 (1,2 done in class)
• Homework 5, Tue 9/20/07 : READ SECTION 1.6, 1.7. We worked on writing cross product and double cross product in index notation. Learned how to remember, or more correctly how to reconstruct formula (44) in the notes. This is the index notation version of the double cross product identity (25), but it is not equal to (25), it's actually more general. We wrote the LHS of (25) in index notation, then used (44) to obtain the RHS of (25) in index notation and finally back to Gibbs/vector notation. We discussed `formula (46)' and its important geometric meaning: signed volume.
  • 1.6. 1--4 (1,2 done in class)
  • Work to ensure that formula (54) is as easy for you as (a+b)<sup>2</sup> = a<sup>2</sup>+ 2ab + b<sup>2</sup> (you know that, right?)
  • Work to ensure that you understand that the LHS of (55) is the RHS of (55) without having to `count on your fingers'.
  • Absorb (56). &epsilon_ijk gives you an explicit formula for determinants that can be straighforwardly adapted to higher dimensions.
  • 1.7. 1--3 Basic
  • 1.7. 7--10 Scary at first, but not too hard once you get the hang of it. You want to get the hang of it.
  • 1.7. 11--12 Same problem in different forms (do you see that?). This is a recipe that you may know as Cramer's rule, here you see where it comes from.
  • 1.7. 13--15. OK, not so basic now, but hey, you're getting good at this, aren't you?
• Homework 6, Tue 10/02/07 : READ SECTION 1.8, 1.9, 1.10. This is about geometric and physical applications of vectors.
  • 1.8. Points, Lines, Planes, etc. Several problems/examples done in class (eqn of lines, planes, distance between point and plane, between 2 lines). Digest those well, they are basic. Try other problems in that section.
  • 1.9. What does r(t) represent, in general? What is the geometric interpretation of dr/dt? Try and digest the 3 problems on page 23. They will come back and bite you. (They come up in 1.10 and were solved in class as part of those 1.10 examples).
  • 1.10. Motion of a particle. First think about solid body rotation (a.k.a. rigid body rotation, uniform rotation), which was discussed at length in class. Look at Motion of a charged particle in a magnetic field , this is closely related to rigid body rotation. You should know linear motion, uniform acceleration, rigid body rotation, and motion due to a central force and be able to deduce Kepler's law.

CHAPTER 2: Vector Calculus.

• Homework 7, Tue 10/16/07 : READ SECTIONS 1.1, 1.2, 1.3.
  • 1.2. 1-6.
    Note: for 1.2.3, there are 6 direction cosines (know what those are?) but you only need 3 angles in general. These are Euler angles which we skipped this Fall 2007. But you can understand them in terms of spherical coordinates which we discussed in class today. Think of e₁ as a point on a sphere of radius 1. That's what it is really. So you can specify it by specifying two angles: its longitude and polar angle with respect to the cartesian basis e_x, e_y, e_z. In general, we can write e₁ = sin &theta cos &phi e_x + sin &theta sin &phi e_y + cos&theta e_z, for some &phi and &theta . So we only need TWO angles to specify e₁. [Exercise: check that |e₁| = 1]. Once you have e₁, you know that e₂ is perpendicular to e₁, so you only need one more angle, a rotation angle about the e₁ axis, to specify where e₂ is. We can write down an explicit formula for e₂. How? First, define 2 unit vectors orthogonal to e₁ and to each other, use cross products to do that, e.g. e_z x e₁ and e₁ x (e_z x e₁) are perpendicular to e₁ and to each other (agree?). Think of these as the appropriate "x" and "y" axes with e₁ pointing in the "z" direction. Now they just need to be normalized to length one, by dividing by the magnitude | e_z x e₁| and we can write e₂ = cos &alpha (e_z x e₁) / | e_z x e₁| + sin &alpha (e₁ x (e_z x e₁) ) / | e_z x e₁| , for some &alpha . This e₂ is the most general form of a unit vector perpendicular to e₁. Substituting the earlier expression for e₁ in terms of e_x, e_y, e_z and grinding it out, you could find an explicit expression for e₂ in terms of &alpha , &phi , &theta and e_x, e_y, e_z. [Exercise: Explain/Show why |e_z x e₁| = sin &theta ].
  • 1.2.4 extra: Approximate the length of the ellipse of major axis a, minor axis b using both (i) a cartesian parametrization y=y(x), and (ii) the &theta parametrization given in 1.1.4. Focus on 1/4 of the ellipse by symmetry, renormalize the problem to remove a and/or b from the integrals if possible. Is there a single formula for the perimeter of the ellipse in terms of a and/or b? (say something like &rho (a+b) , with &rho some universal constant, just like the perimeter of a circle of radius r is 2 &pi r). Pick a/b = 2. Use N=2,4,8,... segment approximations with equi-spacing in x for (i) and &theta for (ii). Sketch the broken line approximations to the elliptical arc used in both (i) and (ii).
  • Digest spherical coordinates in section 1.3. You can start looking at exercises 1.3.1-5.
• Homework 8, Thur 10/18/07 : READ SECTIONS 1.3, 1.4. We discussed general parametric respresentations of a surface and the special cases of cartesian parametrization and spherical coordinates. Formulated area of northern hemisphere in terms of cartesian and spherical coords. We were greatly relieved to find out that area of Northern hemisphere is indeed 2 &pi R²! (so full spherical area is ...).
  • 1.3. 1 - 5. For 1.4, look at e₁, e₂ info above. No need to push it to all the ugly details. Just want to feel like you have a good understanding of what's involved and going on.
  • 1.4. 1 - 5. #5 may be hardest, so don't attempt this one until you've got the other ones under control.
• Homework 9, Tue 10/23/07 : READ SECTIONS 1.5, 1.6 We discussed spherical coords and general mappings r = r(u,v,w), need to understand coordinate curves, coordinate surfaces, meaning of partial derivatives of r with respect to u, v, w , how to make line elements, surface elements and volume elements.
  • 1.5. 1--4. 1, 2, 4 were basically done in class. Make sure you can do them on your own.
  • 1.6. 2, 3, 4.
  • [`Advanced' but cool and practical, Mercator had to figure this out in 1569! and Google maps people also!]
    Consider r = r(u,v). You know, or should know, what orthogonal coordinates means. There is another important concept in mappings called conformal coordinates . Conformal coordinates are coordinates that preserve all angles , that is, angles between any two lines in the (u,v) plane are the same as the angles between the corresponding curves in r-space (a line in the u,v plane is a curve in r-space, right?). What do the partials of r with respect to u, v need to satisfy in order for the coordinates to be conformal? Are conformal coords orthogonal? Are orthogonal coords conformal? Are longitude-latitude conformal coords for a sphere? Are polar coords conformal?
• Homework 10, Tue 10/30/07 : READ SECTIONS 1.7, 2.1--4
  • 1.7. 1-6. And make sure you understand the basic Carnot cycle P, V example done in class.
  • 2.1 Bottom of page 15, and more, solved in class.
  • 2.4 Must know (i.e. be able to quickly rederive/reconstruct) all basic identities and be able to prove them using indicial notation. We proved (65), curl of gradient, in class. Prove (64), div of curl. Try out all exercises, pay particular attention to #7, 8, 9.
• Homework 11, Thur 11/01/07 : READ SECTIONS 1.7, 2.1--4
  • 2.4 Let w be a constant vector and r be our good old position vector.
    1. Calculate grad f, where f=f(r) = A + B r . w , A, B constants, using both Gibbs and index notation.
    2. Calculate (w . grad) r using both Gibbs notation (done in class, remember the meaning of directional derivative) and index notation.
    3. Calculate curl v where v= w x r (this was done in class).
    4. Calculate curl r , really quickly (how?! look back at bottom of page 15 in the notes).
    5. Use the previous two results to show that v . curl v is not zero in general.
• Homework 12, Thur 11/08/07 : READ SECTIONS 3.1--3.7: (you may skip proofs of Stokes theorem on pages 22-24, although understanding the proofs is proof that you understand all these concepts.)
  • Do you know and understand the Fundamental Theorem of Calculus in 1D? Could you prove it?
  • Do you understand its generalizations to 2D and 3D? Do you understand Stokes theorem? What are the various ingredients in that theorem: what curve, what surface, what orientations, what vector field?
  • 3.7. 1--11.

CHAPTER 3: Complex Variable Calculus.

• Homework 13, Thur 11/15/07/Tue 11/20/2007 : Read, I mean study, section 1 (pages 1--9). You should be familiar with basic complex algebra: how to add, subtract, multiply and divide complex numbers. What is the norm and conjugate of a complex number? What are the cartesian and polar representations of complex numbers? What is the exponential, sine, cosine and log of a complex number? What is Euler's formula? What are e ³⁺²ⁱ, cos(3+2i) , sin(3+2i) , ln(3+2i) ? What is ln(-2) , were you not told before that ln(-2) did not make sense?! Should you ask for your money back? What could iⁱ and iⁱⁱ be? What is (2+3i) ²⁺³ⁱ ? Test your understanding by trying as many of the exercises as possible. If stuck, go back to the notes and try to learn what you are missing.
  • 1.1.1, 2
  • 1.2.1, start by proving it for z², z³, ... (that's what you would do first, isn't it? or would you jump to zⁿ rightaway?)
  • 1.3.1,2
  • 1.4.1, 2 (do you really understand the ratio test, or is it just a recipe to you?), 1.4.3-7 are along the same vein as formula (16)-(19), do you get the idea?
  • 1.5.1--7. Try them all. (7) is `hardest'. (5), (6) and (7) tell you how to do it, do you understand the hints? Do you understand the questions?
  • 1.7.1-3. We did several examples of #1(ii) and #2 in class.
  • The Alex Richetta/John Kenney exam problem: Prove that if all the coefficients c₀, ..., c_n in (47) are real (that's what people mean by a "real" polynomial), then if z is a root, so is z*.
• Homework 14, Tue 11/27/07 : Section 2. You should understand the mapping w = z² inside out (section 2.2 example).
  • Could you reconstruct that sect. 2.2 picture, all by yourself, under exam/time pressure?! (without looking at answer)
  • Could you construct the equivalent picture for w = z³ ? What about w = e^z and w = ln(z) ?
  • For w = z² , we showed in class that the vertical lines u=const in the w-plane correspond to hyperbolas in the z-plane, and horizontal lines v=const also correspond to hyperbolas. Can you construct the reverse mapping, i.e. what do the vertical lines x=constant in the z-plane correspond to in the w-plane? What about the horizontals y=constant?
  • 2.2.1--4. All good stuff. Don't worry about cosh(z) but have a look at 1/z, exp(z) and Joukowski, the latter is a basic, famous mapping used in aerodynamics 101.
• Homework 15, Tue 12/04/07 : Section 3.
  • What is a complex integral? Does the definition (53) make sense to you? What about formula (59)? Do you understand the examples done in class? How about the 3 examples on page 14?
  • What the heck is going in equation (60)?! How about (61)?
  • You must know and understand (63), (64).
  • Read and digest the discussion of ln z on page 16. It's tied to exercises 6 and 7. Try to understand what is going on here, not look for a quick plug-and-chug formula to memorize.
  • Try out the exercises after having tried to read and understand the concepts of pages 14 and 15. Several of the exercises are essentially solved in the text.