CHAPTER 1: Vectors.

• Homework 1, Tue 9/2/08 : STUDY SECTIONS 1.1, 1.2
  • Visualize/sketch the 8 properties (1)--(8) of vector additions and multiplication by a real number.
  • Define linear independence , basis , components
  • Are coordinates and components the same thing?
  • An airplane travels at airspeed V, heading &theta (measured clockwise from from magnetic north). Weather reports state that the wind has speed W heading &phi . Derive the formula for the airplane's groundspeed (in magnitude, direction form).
  • 1.2. 1, 5--8
• Homework 2, Thur 9/4/08 : STUDY SECTIONS 1.3, 1.4. you are responsible for all of section 1.4. You *can* skip definitions of dot product and norms in Rⁿ on pages 9, 10. Make sure you digest formula (14) and (15) they come up over and over and over.
  • 1.3. 1--7,9, do 1.3.4 two ways: geometrically and vector-algebraically
  • 1.4. 1--5.
• Homework 3, Tue 9/9/08 : STUDY SECTIONS 1.4, 1.5. Make sure you are comfortable with index notation.
  • 1.4. 1--5. (Yeah, I said that already, but did you do them or watch football all weekend?)
  • From the geometric definition of the cross product, show that the distributivity property a x ( b + c ) = a x b + a x c holds.
  • 1.5. 1, 2. 3 geometrically only.
• Homework 4, Tue 9/11/08 : Keep STUDYING SECTIONS 1.4, 1.5!
  • We derived in class what a x ( b x a ) is, using our geometric understanding of cross product. What about (a x b ) x a ? can you figure it out as we did?
  • We solved exercise 1.5.3 geometrically, can you? How about vector algebraically now, how does that work?
  • 1.5. 4, 5 (basically done in class, right?), 6, 8 (lots of algebra, but nowhere near as much as if you blindly used the determinant recipe (top of page 13) to compute the 2 cross products).
  • Show (25) (first form say) by writing c = &alpha a + &beta b_perp + &gamma a x b, where b_perp is the component of b perpendicular to a. What is &alpha =? What is &beta = ? Why don't we care what &gamma is? How does that help us to figure out (25)? [Hint: remember that we figured out what (a x b ) x a is]
• Homework 5, Tue 9/23/08 : Sections 1.6, 1.7
  • 1.6. 1--4.
  • 1.7. 1, 2, 3, 5,6, 7--10 (all related), 11, 12. [optional] IF all these exercises are pretty easy for you, try out 15 and 16 also. Notice the form of the formulas in 10 and 16, these can be extended to higher dimensional spaces.
  • What is the area of the triangle whose vertices have the cartesian coordinates (x₁ , y₁ , z₁) , (x₂ , y₂ , z₂) , (x₃ , y₃ , z₃) ?
  • What is the volume of the tetrahedron whose vertices have the cartesian coordinates (x₁ , y₁ , z₁) , (x₂ , y₂ , z₂) , (x₃ , y₃ , z₃), (x₄ , y₄ , z₄)?
• Homework 6, Thur 9/25/08 : Sections 1.8. 1.13
  • We discussed exercises 1.7.7 in quite a few details today and reviewed index notation and summation convention. Review all that well.
  • Try out all exercises in section 1.8 (not numbered). We also briefly discussed cartesian coordinates and reviewed how to go from vector equations to cartesian equations (and vice-versa). Are you comfortable with that?
  • For all the following problems: solve in coordinate-free form first, then translate into cartesian coords:
  • What is the distance between the point (x_P ,y_P ,z_P) and the plane A (x-x₁) + B (y-y₁) + C (z-z₁) = 0 ?
  • What is the distance between the point (x_P ,y_P ,z_P) and the plane A x + B y + C z= D ?
  • What is the distance between the point (x_P ,y_P ,z_P) and the line (x-x₁)/A= (y-y₁)/B= (z-z₁)/C ?
  • What is the distance between A₁ (x-x₁)= B₁ (y-y₁) = C₁ (z-z₁) and A₂ (x-x₂)= B₂ (y-y₂) = C₂ (z-z₂) ?
• Homework 7, Tue 9/30/08 : Sections 1.9, 1.10
  • Exercise 1.7.11 discussed "at length" in class!!!
  • Google search "projection". What does "projecting a point onto a line or a plane" mean (several possible meanings but what's the usual one)? What does "projecting a vector onto another vector" mean?
  • Section 1.9 all exercises (not numbered)
  • Section 1.10: What is Newton's law? What is free motion?
  • Section 1.10: Consider (70).
    1. Explain/show that the motion is in a plane. What plane? Is the motion a plane or a curve?!
    2. (70) is a parabola. Can you verify that claim by writing it in the good old form y = a x² + b x + c for some constants a, b, c? What are those directions x and y? What are the constants a, b, c in terms of the vectors r₀, v₀, a₀? Which point is the "bottom of the parabola" in terms of r₀, v₀, a₀?
  • Section 1.10: Consider the vector differential equation da/dt = c x a, where c is a constant vector. Prove that the magnitude of a is constant. Prove that the vector projection of a onto c is constant. What does that tell you about a(t)? Can you make a "side view" and a "top view" of a(t)?
• Homework 8, Thur 10/2/08 : Section 1.10
  • Everything discussed in class is covered in these extra notes . We did the "direct vector approach" (page 2) but also reminded you of what happens when a particle goes around a circle at constant speed (that's equations (6). Equations (6) also correspond to the harmonic oscillator , very basic in physics and engineering. )
  • Pick you favorite cartesian axes x, y, z . Consider the vector (3,2,1) in those axes. We want to rotate it about the direction (1,2,3) by &pi/3. What are the coordinates of the rotated vector? Explain in detail how to go about doing that. Better yet, deduce a general formula for the rotation of (v1,v2,v3) about (a1,a2,a3) by angle &theta .
  • Show that if a particle moves under the action of a central force then the motion is (1) planar (what plane?), (2) the radius vector sweeps equal areas in equal times, (3) the motion conserves an `energy' that consists of the kinetic energy + a `potential energy'. Don't regurgitate, that's foul! Can you explain all this to a smart friend? (you do have a least one smart friend, don't you?)

CHAPTER 2: Vector Calculus

• Homework 9, Tue 10/7/08 : Sections 1.1, 1,2
  • We discussed various problems from HWK 6 and section 1.8.
  • 1.2. 1-3 basically solved in class today.
  • Ok, so we can write our good old ellipse, centered at O and aligned with the x, y axes in the implicit form (x/a)² + (y/b)² = 1 , OR we can write it in the explicit parametric form x = a cos &theta , y=b sin &theta . What is that &theta ? is it the polar angle (the angle between the x axis and the radial line from O to the point (x,y))?
  • Elliptical fun: A more geometrical, and practical, definition of an ellipse is that it is the set of points for which the sum of the distances to two distinct points, called the foci (one focus, two foci), is a constant.
    • Explain why that constant length = 2 a, where a is the major radius. (try to understand geometrically what is going on by making sketches and visualizing what 2 a is).
    • If the foci are F₁ and F₂, and the length is 2 a, provide an implicit vector equation for a point P on the ellipse. What is the minor radius b and where is the center of that ellipse (in terms of F₁, F₂, a)? For the canonical equation of the ellipse, x²/ a² + y²/b² = 1 , where are the foci?
    • Assume you are free to pick your x and y axes so that the foci are at (-c,0) and (c,0) and the sum of the lengths is 2 a, (and the ellipse is in the (x,y) plane), deduce that the equation of that ellipse indeed has the form x²/ a² + y²/b² = 1 for some b. Express b in terms of a and c.
    • Why do your parents, teachers and coaches tell you to "focus" ? (answer is related to the following problem:)
    • Prove that the angles between the line F₁ P and the tangent to the ellipse at P is equal to the angle between the tangent and the line F₂ P. ("easy" using vectors, parametrization and geometric definition of ellipse, but requires rather good conceptual understanding!)
  • Homework 10, Thur 10/9/08 : Sections 1.1, 1.2
    • 1.2. 4, 5 done in class. 6 (Physics connection: think of F as a B, the magnetic field due to a line current passing through O and flowing in the z direction.)
    • Next Tuesday, we'll cover BOTH 1.3 and 1.4 and that we'll be part of the exam material!! so you may want to look at that now already.
  • Homework 11, Tue 10/14/08 : Section 1.3
    1.3. 1 (done in class) and 2.
  • Homework 12, Tue 10/21/08 : Section 1.3: discussed spherical coordinates, following physics conventions
    • 1.3. 1 (done in class), 2 (done in class), 3, 4, 5
    • Define spherical coordinates for a 4-dimensional sphere: x₁²+x₂²+ x₃²+x₄² = R² .
  • Homework 13, Tue 10/28/08 : Sections 1.3-1.8: discussed spherical coordinates, cylindrical coordinates, curvilinear coordinates, orthogonal coordinates, mappings, change of variables, Jacobians. As always, don't be shy about going back to your favorite elementary calculus book to remind yourself of all these concepts.
    • 1.4. 1--5. (#1, #3(i),(ii), #4, #5 done in class!)
    • 1.5. 1--4. (#1 do it for (15) only, basically done in class)
    • 1.6. 1--4 (#1 done in class except for end of the problem)
    • 1.7. 1, 2, 4, 5, 6 (#1 started in class, you finish it. Study sections, 1.7.1, 1.7.2 for a very useful property of Jacobians: eqn (45). The key mathematical steps are (39), (40) & (41) [mostly notation], (43)--(45). Note that 1.7.1, 1.7.2 completely solves another problem: calculating work done by a gas during a Carnot cycle. That's an "area" problem but in the P-V "plane" (where P is pressure and V volume of a gas. "Area" in P-V plane is actually "work" (check: Pressure = Force/Area, so Pressure x Volume = Force x length = Work, OK). That `area' calculation is done in a couple of lines: equations (39), (48) and (49) ).
  • Homework 14, Tue 11/4/08 : Section 2.1-2.4: Grad, Div, Curl remember Extra notes on gradient, gradient in spherical coords that were covered in class.
    • What is the gradient of f=f(r) where r is distance to the origin and why? Done in class Thur 11/6/08 !!
    • What is the gradient of f=f(s) where s is distance to point C and why? Done in class and in notes!!
    • If F(r) = F(r) e<sub>r</sub> (cf. bold = vector) can we always write F = grad V(r) ? Explain. Done in class Thur 11/6/08 !!
    • Derive the expression for the del operator in cylindrical coords.
    • Prove vector identities (64)--(68), (70), (71) using index notation. Learn to rebuild them quickly using vector notation
    • 2.4. 5--9. #7, #8 Done in class Thur 11/6/08 !!
    • Calculate the divergence and the curl of f(r) e<sub>r</sub> (central force!). [Hint: use results of earlier exercises] Done in class Thur 11/6/08 !!
    • Calculate the divergence and the curl of f(&rho) e<sub>z</sub> where &rho is distance to the z axis (general current flowing in z direction!). [Hint: use vector identities and your understanding of gradient]. (related, but not identical, to #9)
  • Homework 15, Thur 11/6/08 : Sections 3.1-3.3: FTC, FTC2 v1.0, FTC2 v2.0
    • What is the fundamental theorem of Calculus? Explain/show why it is true.
    • C(x,y) is the amount of algae per unit area in lake Mendota. Someone measured C<sub>n</sub>= C(x<sub>n</sub>, y<sub>n</sub>) at a bunch of discrete locations (x<sub>n</sub>, y<sub>n</sub>) , n=1,... ,N spread out fairly uniformly over Lake Mendota (OK, if you really want to know N=753). Your task: estimate the total amount of algae in lake Mendota. Explain in a few words/math expressions how you would go about doing that.
    • Prove (81).
    • Prove (83) ab initio (what does that mean?!?)
    • Prove (83) from (81).
  • Homework 16, Thur 11/13/08 : Sections 3.1-3.7:
    • If A is the area inside the curve defined in exercise 1.2.2, write the right hand side of (84) as an integral over &theta (complete with limits of integration and explicit dependence of F and G on &theta . Likewise, write the right hand sides of (105) and (106) as &theta integrals.
    • How could you use (84) to calculate the area A inside the closed curve, knowing that closed curve? Be explicit provide 2 distinct line integrals that give the Area. What would Kepler have to say about this?
    • Digest Stokes (86) and Divergence theorem (111). You are expected to know those theorems and be able to apply them.
    • Digest the simplest and more general form of the `divergence theorem' (113), and be able to deduce the many forms of the divergence theorem from it.
    • 3.7 1 (done in class) , 2 (basically (done in class) but ...), 3 (done in class) (at length, fundamental problem to understand the recipes you use in E&M), 4 (done in class) , 5 (read section 3.5), 6, 7 (done in class) , 8, 9, 10 (basically (done in class) ). (If you're an AMEP or Physics major, you probably want to understand 11-12, good stuff!)
    • A volume is specified by its surface boundary S (that is, you've been given a lot of data points that specify S, you can build up lots of little triangles on S to calculate any integral over S, right?). How can you compute the enclosed volume?

  CHAPTER 3: Complex Calculus

  Numbered exercises refer to the Lecture notes on Complex Calculus
  • Homework 17, Thur 11/20/08 : Sections 1.1--1.5
    • 1.1. 1--3
    • 1.3. 1, 2
    • 1.4. What is the ratio test? WHY is it true? Consider the series 1 + a + (a/2)² + (a/6)³ + (a/24)⁴+ ... +(a/n!)ⁿ+... . Does it converge? For what complex a?
    • 1.5. 1---7
  • Homework 18, Tue 11/25/08 : Sections 1.1--1.6: Geometric and Taylor series applications
    • Find the Taylor Series of f(x)=1/(1+x²) about x=0 . Plot f(x) together with several partial sums of the series for -3 < x < 3 .
    • Find the Taylor Series of f(x)=1/(1+x²) about x=1 . What mistake did I make in class? Plot f(x) together with several partial sums of the series for -3 < x < 3. Could you find a closed form formula for the Taylor series about any real number a ? [OK, this is A⁺ material I guess]
    • In high school algebra, you learned a formula for (a+b)ⁿ where n is a positive integer. What is that formula? Do you believe it? Convince yourself and me that it is correct.
    • We defined exp(z) = 1 + z + z²/2! + z³/3! + z⁴/4! + ... in class. This series works for any complex z as we showed using the ratio test. Show that exp(z+a) = exp(z) exp(a) from the series definition. That allows us to write exp(z) = e^z. Ask Matthew Warns about e .
    • Use series to find a function f(z) such that f(0) = 1, f'(0) = 0 and f''(z) = f(z)
    • Use series to find a function f(z) such that f(0) = 0, f'(0) = 1 and f''(z) = f(z)
    • Use series to find a function f(z) such that f(0) = 1, f'(0) = 0 and f''(z) = -f(z)
    • Use series to find a function f(z) such that f(0) = 0, f'(0) = 1 and f''(z) = -f(z)
    • Use series to find a function f(z) such that f(0) = 1, f'(0) = 0 and f''(z) = z f(z)
    • Use series to define cos z and sin z. Prove eqns (31) and (32) in the notes.
    • 1.6. 1--6.
    • Store in permanent memory: Geometric sum and series, general formula for Taylor Series about arbitrary point a, Taylor series about z=0 for e^z, cos z, sin z , Euler's formula e ^{i z} = cos z + i sin z , cos z = (e ^{i z} + e ^{-i z} )/2 , sin z = (e ^{i z} - e ^{-i z} )/ (2 i) , ln z =?
  • Homework 19, Tue 12/02/08 : Sections 1.6--1.8
    • If z = x + i y as usual, express | e^z| and arg(e^z) in terms of x and y , what about | ln z| and arg(ln z) ?
    • Show that (cos z)² + (sin z)² = 1 for any complex z .
    • Show that arccos z = - i ln [ z +/- sqrt(z²-1) ] (where arccos z is the inverse cos, sometime written cos^-1 which is confusing since we write cos² z to mean the square of cos z).
    • If you got the previous problem, you should have no problem with
      1. arcsin z = - i ln [ iz +/- sqrt(1-z²) ]
      2. arctan z = (i/2) ln [ (i+z)/(i-z) ]
      3. arccosh z = ln [ z + sqrt(z²-1) ]
    • What are all the possible values of 3ⁱ ? Show that i³ has only one possible value, and it's the one it should be. What about i^3.21?
    • 1.8. 1, 2, 3
  • Homework 20, Tue 12/09/08 : Sections 2.1-2.2, 3
    • 2.1. 1, 2
    • 2.2. 1, check out complex map examples .
    • 3 Calculate the integral of z² and z^-1 along the path z = t + i (1-t²) for t = 0 to 1 .
  • Homework 21, Tue 12/11/08 : Section 3
    • Must know and be able to justify Cauchy's theorem (i.e. explain why it is true to an intelligent skeptic who does NOT just believe everything you say)
    • Must know and be able to justify all that goes into computing the simple integral (67)
    • 3.1. 1, 2, 3 (You are responsible for partial fractions, remember Math 222).
    • 3.2. Must fully understand complex integral approach to calculating (84).