15 Homework sets on CHAPTER 1: Vectors.

CHAPTER 2: Vector Calculus

• Homework 16, Wed. 2/25/09 : STUDY SECTIONS 1.1, 1.2
  ACHTUNG!! THIS IS MATH 222 material again for the most part !!
  Did you get rid of your Calc book?! aie, aie, aie! Lucky for you Prof. Angenent has Free Math 221 and 222 lecture notes . He uses x for the position vector while we use r (for "radial vector") like physicists typically do, but otherwise, it's all there at the elementary Calc level: lines, planes, distance, dot and cross, curves, tangents, arclength... bold vectors as well as cartesian forms,...
  1. What does r= r(t) represent, where "t" is a real number in a certain range, say t₀ ≤ t ≤ t₁?
  2. What does dr/dt represent? What about T=(dr/dt) / |dr/dt|, what's that?
  3. What about dT/dt? What is that in terms of derivatives of r(t)? Why would we care about dT/dt? What is that telling us about the curve? What kind of curve has dT/dt=0 for all t? (yes, yes, you saw all this in Math 222).
  4. Given r= r(t), what is the equation of the tangent to the curve at the point r_A= r(t_A)? What would be the form of that equation in cartesian coordinates?
  5. What is the distance between point A and point B along the curve r= r(t)? That is, you are driving from Madison to Chicago along highway 90, a known r= r(t) curve, not `as the bird flies' . What would all that look like in cartesian coordinates? What would it look like if the curve is given in Math 221 form as y=f(x)? What if it's a curve in 3D space (don't we live in 3D?) specified as y=f(x), z=g(x)? (Wait! is that one curve or two curves?!). What if the curve is specified as x=x(t), y=y(t), z=z(t)? (yes, yes, you saw all this in Math 222).
  6. What is the parametric equation of a circle? Write its vector AND cartesian forms.
  7. What is the parametric equation of a circle of radius R centered at point C and perpendicular to the direction n?
     • What are the vector and cartesian parametric equations of a circle of radius 5.123 centered at point (0.5,1.2,4) and perpendicular to the direction (8,5,6)?
       WARNING!! If you want to get big brain muscles: don't look at the solution too quickly... Too late, you did look, didn't you? Done in class 2/25
     • What are the vector and cartesian parametric equations of a circle of radius R centered at point (c₁, c₂, c₃) and perpendicular to the direction (n₁, n₂, n₃ )? Which, really, is the same as: What is the parametric equation of a circle of radius R centered at point C and perpendicular to the direction n, isn't it?
       OK, since it looks like we're going around in circles about circles:
  8. You remember that x²+y²=R² is the implicit equation of a special circle (which one?), but what would the implicit equation look like if the circle was centered at some arbitrary point C and perpendicular to the direction n ? (No, not again!?! it's that SAME problem again! basically, although now we want the implicit equation, not the explicit parametric equation)
  9. Vector Calc Section 1.2. 1, 2, 3 (ask your favorite instructor about "How many independent angles? Specify those angles")
  10. Find the simplest vector parametric equation for the curve that goes through points P1 and P2.
  11. Find the simplest vector parametric equation for the curve that goes through points P1 and P2 and is parallel to a at P1. (hint: use products of (1-t) and t as needed with t=0 → 1. Go back to previous problem first).
  12. A: Find the simplest vector parametric equation for the curve that goes through points P1 and P2 and is parallel to a at P1 and to b at P2. (hint: use products of (1-t) and t as needed with t=0 → 1). Got to do previous problem before this one.
  13. A: Find the vector equation for the circle that passes through points P1, P2 and P3. Is that circle unique? Can you write a Matlab/Octave code to find that circle? (1) Find center and radius, (2) parametrize circle (that's #7 over again).
• Homework 17, Fri. 2/25/09 : STUDY SECTIONS 1.1, 1.2
  • 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 θ , y=b sin θ . What is that θ ? is it the polar angle (the angle between the x axis and the radial line from O to the point (x,y))? Done in class

    WARNING!: in physics, you are likely to study motions of planets. Kepler's 1st law is that the planets move along ellipses with the sun at one focus. To deduce this from Newton's law people generally work in polar coordinates with the origin at one focus ("heliocentric" coordinates). The θ that appears there is a polar θ, not the same as the θ that we use in our simple parametrization. See e.g. Mathematics of the ellipse and derivation from Newton's law .

  • Elliptical fun: A more geometrical definition of an ellipse is that it is the set of points for which the sum of the distances to two distinct points, F₁ and F₂, 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). Done in class
    • 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. Done in class What is the minor radius b and where is the center of that ellipse (in terms of F₁, F₂, a)? For the canonical (or standard) equation of the ellipse, x²/ a² + y²/b² = 1 , where are the foci? Done in class
    • 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" ? Done in class (answer is related to the following problem:)
    • Prove that the angle 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 as |F₁P| + |F₂P| = 2a, but requires rather good conceptual understanding!) Begun/sketched in class. Hint: F₁ and F₂ are fixed but not P. The point P=P(t) for some parameter t. We really don't need to know P(t), i.e r(t), explicitly here, just that there exists such a P(t). Since you'll have to take derivatives of magnitude of vectors, you better make sure you're OK with Chapter 1, HWK 12, #5 first.
• Homework 18, Mon. 3/2/09 : STUDY SECTIONS 1.1, 1.2 Remember line integrals? (yes, yes, you saw all this in Math 222 and 234).
  • You are responsible for understanding integrals (3)--(7) in the vector calc notes. We saw (7) when we discussed Kepler's law, and you had basically seen it when you helped out Farmer Joe, didn't you? Do you see the connection with Farmer Joe?
  • 1.2. 4,5,6 (parts of #5 and #6 were done in full details in class, and they'll be done again in your physics courses)
  • Figure out the sum (integral) over a curve C from point A to point B of F.dr where F is the inverse square law force (gravity force, Coulomb force). Yes you know what I mean. Use the same approach as we did in class for ∫_C r.dr, wasn't that a cool approach? You should be able to handle this if you understood that calculation and our earlier derivation of conservation of energy in the Kepler/planetary motion problem. Look back at your notes on that. Basically done in class on Wed. 3/4 as a reply to Svyat(oslav). We reduce these vector line integrals to a single integral over the single real parameter "t" whatever it is and therefore to a math 221 integral, but then we need to know and have understood the Fundamental Theorem of Calculus AND the chain rule!. The beauty of the whole thing is that we end up calculating this class of line integrals without ever having to know r(t) explicitly. That's only for this special class of central forces F(r)=F(r) r/r however, not true in general. Still works for any F(r)! Pretty forceful...
• Homework 19, Wed. 3/4/09 : STUDY sections 1.3, 1.4. Surfaces, parametrization, tangent vectorS, surface element, area.
  • 1.3. 1,2,3,4,5. Don't push #4 too far, just try to get an idea of what's going on. Same business as in 1.2.2 and 1.2.3.
  • 1.4. 1,2,3,4. Try to understand why the total pressure force on a surface S is ∫_S p dS where p=p(r) is the pressure at point r (p is a scalar field) and dS is the surface element. The sum (integral) must be done over all the surface. Think of little triangles again.
• Homework 20, Fri. 3/6/09 : STUDY sections 1.3, 1.4, 1.5. We discussed cartesian, cylindrical and spherical coordinates. Took both an algebraic approach (i.e. Pythagoras a go go) and a geometric approach. Hopefully you wrote it all down because the notes are slim on all that at this time, but it's basically all here in this pretty good page. NOTE that we follow ISO 31-11, not "american mathematicians". Note also that that page uses both the "hat" notation AND the e_r, e_θ , e_φ notation for unit basis vectors (way down at the bottom of the page). SVYAT: do you see what del is? . So we can say "partial" instead of the shorter "d" but we cannot say "del". That's taken for something important.
  • Position vector r in cartesian, cylindrical and spherical
  • Spherical: longitude+latitude or longitude+polar angle, what are the differences?
  • "line elements" in the coordinate directions, what's a `line element' anyway?! calculations using a `hybrid approach'. Direct calculations using rotations! (we'll get back to that).
  • orthogonal coords? Surface element?
  • Brilliant point made by Wes: range of angles, usually we pick 0 ≤ φ < 2 π, 0 ≤ θ &le π but why not 0 ≤ φ < π, 0 ≤ θ &le 2 π?!
    Problem 1: we'd need an absolute value of |sin θ| in the surface element.
    Problem 2: when moving around the earth around a parallel we'd suddenly switch from coordinates φ, θ to coordinates φ-π, 2π-θ. (example start at φ=0, θ=π/4, and move east increasing φ but keeping θ fixed. What happens to the coordinates when you cross φ=π?).
    So what do you think: standard spherical coords, or Wes-pherical coords?
• Homework 21, Fri. 3/13/09 : DIGEST sections 1.5, 1.6, 1.7. We discussed parameterizations of curves, surfaces and volumes. Discussed spherical coordinates some more, this time focusing on the VOLUME ELEMENT dV. We solved exercise 1.6.1 for spherical coordinates. Discussed JACOBIAN DETERMINANTS (1.7 eqns (33), (34), (37), (38)), discussed their reciprocity property (section 1.7.2) and the general way to change variables in multiple integrals (section 1.7). We posed the Carnot cycle example (sect 1.7.1 and 1.7.2), except that we just focused on the math using x, y and u, v instead of the physical P, V and S,T.
  For those of you who, like John WIETING, were on spring break already, you may have to purchase notes from someone who was at the lecture! (yes, some students were there!
  • You should study sections 1.5, 1.6, 1.7 of the lecture notes.
  • You could check out wikipedia pages on Jacobian and on Change of variables in multiple integrals
  • You could
    • watch this Berkeley lecture on change of variables and Jacobians
    • and this one on more change of variables and Jacobians (solves one of our exercises!).
    • or this MIT lecture on change of variables and Jacobians (also solves same exercise).
      [All of these movies are at the lower 234 level than the 321 level, and they are for Berkeley/or MIT students. Not quite as good as Math 321 UW students! but do you see the nice boards and chalk they have!?]
  • 1.7 Carnot cycle example discussed at length in the notes + #1, 2, 4, 5, 6. If you have trouble with these then you need to watch/study this MIT lecture on change of variables and Jacobians .
  • Jokes or no jokes? I learned today that jokes in class are ``distracting" and some would prefer fewer jokes in class. Let me know! (I'm not joking).
• Homework 22, Mon. 3/23/09 : Section 2.1, 2.2, 2.3: Grad, div, curl. Yep, you've seen this before, so what's a gradient, a divergence and a curl? We're going to emphasize the very important geometric properties of the gradient and use speedy vector and index notation calculations when we can avoid the cumbersome cartesian calculations.
  • Remind yourself of what is a gradient. Study your calc 234 book/notes or Watch this MIT lecture on gradient and directional derivatives (complete with thick French accent! This is from the 2nd calc course at MIT, they do Math 221, 222 and 234 in 2 courses there.).
  • TWO sets of notes on gradients for ONE price! aren't you lucky!?: Section 2 of vectorcalc notes and Extra notes on gradient (provides a swift derivation of gradient in spherical coordinates)
  • Know how to compute gradients of spherical and cylindrical functions, i.e. f=f(r) and f=f(ρ) respectively where ρ² = x² +y² and r² = ρ² + z², without having to go through cartesian.
  • Consider the surface (x/a)²+ (y/b)² +(z/c)² = 2 . What kind of a surface is that? If P₀ ≡ (x₀,y₀,z₀) is a point on that surface, what is the equation of the tangent plane to that surface at that point P₀?
  • Know gradient, divergence and curl in (cartesian) index notation.
  • Know how to (re-)derive gradient operator in spherical and cylindrical coords. (Spherical spelled out in Extra notes on gradient . Cylindrical is similar but easier.)
  • A⁺/Honors: Know how to derive div and curl in spherical and cylindrical coords (must study Extra notes on gradient for that. )
  • And now you must all know what ∇ is.
  • in principle, you should be able to derive any formula listed here .
• Homework 23, Wed. 3/25/09 : Section 2.4: vector identities
  • Must learn to rederive them quickly using BOTH Gibbs notation AND index notation.
  • 2.4. ALL 1, (2 and 3 should be old hat for you), 4, 5, 6, 7, 8, 9
• Homework 24, Fri. 3/27/09 : Section 2.4: vector identities CATCH UP!
• Homework 25, Mon. 3/30/09 : Section 3: Fundamental Theorem of Calculus (combined with next homework)
• Homework 26, Wed. 4/1/09 : Section 3: curl form of Green's theorem
  • FTC: Fundamental Theorem of Calculus, what?, why? Must understand (75)--(79). (Math 221 material reminder)
  • What does FTC tell us for ∫_A ∂F/∂x dA? Must understand (80) to (81).
  • What does FTC tell us for ∫_A ∂F/∂y dA? Must understand (82) to (83), AND how to obtain (83) directly from (81).
  • Understand how (81) and (83) can be combined into the single (84): the curl form of Green's theorem.
  • Calculate ∫_C y dx - x dy using both a direct calculation of the integral and theorem (84) when C is (i) the rectangle with lower left corner (x1,y1) and upper right corner (x2,y2), (ii) the circle of radius R centered at the origin. What is the value of the integral for an arbitrary closed curve? (not necessarily plane). What is the connection with Farmer Joe's fence? (homework 5)
• Homework 27, Fri. 4/3/09 : Section 3: Stokes' Theorem
  • Absorb (86). What is S? What is C? how are they related? What is closed, the surface or the curve? What is v?
  • Give at least two acceptable S's if C is the equator of a sphere oriented eastward.
  • Geometric interpretation of the curl: eqn (87).
  • 3.7. 1, 2, 3, 4 (One method for #1 and #2 done before (homework 18), #3 done in class, see below).
  • Prove Nathaniel's theorem: ∫_S ∇ × v • dS=0 for any closed surface S.
• Homework 28, Mon. 4/6/09 : Section 3: circulation of magnetic field (magnetostatics) Exercise 3.7.3.
  • Deduce the expression for the magnetic field induced by a line current B = C (e_z × r)/ &rho², where ρ= |e_z × r| = (x² + y²)^½ and C is a constant, from the Biot-Savart law integrated over an infinite straight current (z axis), without computing any integral! OK, since Svyat asked here it is , but can you explain what's going on in equations (4) and (5) of those notes?!?
  • Calculated curl B ≡ ∇ × B in two ways:
    1. cartesian coords is nice here since B = B(x,y) = C (-y e_x + x e_y) / (x²+y²), where C is a constant. Since B lies in the x,y plane and depends only on x,y, its curl is only in the z direction (prove the latter statement),
    2. cylindrical coords is not bad either since that same B = B(ρ,φ) = C e_φ/ρ, where C is that same constant, but more involved conceptually: need to know/rebuild ∇ in cylindrical coords, see e.g. Del in cylindrical and spherical , then need to know how to compute ∇ × (e_φ/ρ) (done in class).
  • Circulation of B about an arbitrary loop (closed curve C), three steps:
    1. circulation around a circle centered on z axis (direct calculation, old stuff done earlier, homework 18),
    2. circulation about an arbitrary closed curve using Stokes' theorem. Case 1: loop does not encircle current.
    3. Case 2: loop does encircle current. What do we do then?!?
    4. Conclusion: Generalization to many line currents, effectively deducing Ampère's law.
• Homework 29, Wed. 4/8/09 : Section 3: Divergence/Gauss theorem
  • Study section 3.7, you should know all the various expressions of the `divergence theorem'... and you should not confuse then with Stokes' theorem!
  • Ali's question: why is formula (107) true?! Answer: it's the FTC again, but this time in 3D: we integrate a derivative of a function over a volume V and we get the function over the surface area S bounding the volume V, there are + and - signs and some geometry, see (108) and (109). In 2D, that was just a question of rewriting Green's theorem, that's section 3.5, done in class today (formula (106)).
    As I mentioned in class section 3.5 uses poor notation writing dr = |dr|. Those dr's should be left as |dr| or written ds since d|r| ≠ |dr| (why not??! well since you can think of "d" as "infinitesimal" difference, and you are algebra-pros, you should understand that the difference of magnitudes d|r| is not equal to the magnitude of the difference |dr|.).
  • 3.7. 5--11. #7 solved in class Fri 4/10/09 (Proof of Archimedes' principle).