PART 1. Vector algebra: geometric concepts and index notation

• Homework 1, Thur. 9/3/09 : We discussed the basics of geometric/physical vectors, magnitude, direction, unit vectors. Discussed position/radius vector r and how to express it in cartesian, cylindrical and spherical coordinates.
  • Review and complete your own lecture notes so they'll make sense later on
  • Wikipedia on spherical coordinates (pretty good today... also contains cylindrical)
  • The original book on vector analysis (Gibbs/Wilson)
  • Who was Gibbs? When and where did he live? What did he do in science and math?
  • What is Gibbs notation? How do we indicate magnitude of a vector? direction of a vector?
  • What are the units of a unit vector?
  • What is the length of a unit vector? in what units of length?
  • What is r? How do we express it in cartesian, cylindrical and spherical coordinates? You should be able to derive all the formulas `from scratch'. It's not hard if you understand the coordinates, learn to visualize properly (often by taking the appropriate orthogonal projection onto a plane like the x, y plane or the ρ, z plane), and ... you know your basic trig!
  • Make a simple 3D perspective sketch showing all 3 sets of coordinates without just looking them up in your notes or a book
  • What are the cylindrical and spherical coordinates for the points whose cartesian coordinates are (0,0,1), (0,-1,0), (1,1,1)?
  • If we want a unique set of coordinates for each point in 3D space, what range of values do we need for x, y, z, ρ, r, φ, θ ?

• Homework 2, Tue. 9/8/09 : SECTIONS 1.1, 1.2. Linear Combinations and Bases
  • What is a scalar? We will use the word in both its math and its physics meaning. Can a scalar be a function of time and/or position, or is it always a constant? Give physical examples of a scalar.
  • Visualize (i.e. draw sketches to illustrate/demonstrate) the 8 properties (1)--(8) of vector addition and multiplication by a real number.
  • What is a linear combination of vectors?
  • What does it mean for vectors to be linearly independent?
  • What is a vector basis? in 2D? in 3D?
  • Do basis vectors have to be unit vectors?
  • Do basis vectors have to be orthogonal (perpendicular)?
  • Prove that the medians of a triangle are concurrent (i.e. go through the same point) using (1) analytic geometry, (2) vectors.
  • Use vectors to show that the diagonals of a parallelogram intersect at their midpoint.
  • 3 pts are specified by their cartesian coordinates (1,2,3), (2,3,1), (3,2,1). Provide an explicit expression that delivers any point that is inside the triangle formed by the 3 points. What is the explicit cartesian equation of the line that passes through (2,3,1) and (3,2,1)?
  • If r_A and r_B are the radius vectors of points A and B, respectively, show that (α r_A + β r_B )/(α + β) is (the radius vector of) a point on the AB line for any real α, β with α+β not 0. What about (α r_A + β r_B ), for any real α, β, what would that be? (added Sept 10)
  • Vector notes: study sections 1.1, 1.2. Exercises 1.2. 1, 5, 6, 7, 8 (yes, some of these exercises are related) [in 1.2.1 (i) both α and β are between 0 and 1]

• Homework 3, Thur. 9/10/09 : SECTIONS 1.3, 1.5 Dot and Cross products
  • Vector notes: study sections 1.3, 1.5 (Yes, we did it out of order in class). Make sure you completely digest the concepts in eqn (13) and the accompanying figure. Likewise for eqns (19) and (20).
  • Exercises 1.3. 2, 4, 5 (both geometrically and vector algebraically), 6, 7, 8, 9. (#2 was essentially solved in class)
  • 1.5. Do you understand the geometric proof of property 3 on page 11? Could you reproduce it for a classmate without any notes? (Look Ma, no notes!) (partially done in class. Understanding this proof means good understanding of the cross product).
  • Exercises 1.5. 1, 2, 3 (#3 was partially done in class)
  • What are the dot and cross products good for: (1) from a geometric point of view, (2) from a physics point of view? Give examples.

• Homework 4, Tue. 9/15/09 : SECTIONS 1.5 (double cross product), 1.7 (mixed product, determinant), 1.4 (index and sigma notation). 1.6 (Levi-Civita symbol)
  • 1.5. What is the right way to remember and reconstruct eqn (25)? Is (25) a definition or a derivation?
  • Exercises 1.5. 3 (vector algebraically now, using double cross), 4, 5, 6
  • 1.7. Digest equation (46) and the accompanying picture. Can you rederive all that from your deep knowledge of dot and cross product?
  • Exercises 1.7. 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13 (with vector identities for now, not index notation)
  • 1.4. Woaa! do you understand equations (14), (15) and (16)?
    (You may skip the part of section 1.4 on pages 9 and 10. This is Math 340 material, some of you should know it already, the others should learn it soon.)
  • Exercises 1.4. 1--5. 1.5. verify/derive equation (24).
  • 1.6. Definition (27), digest it. Equation (26) and (27), verify them. Study section 1.6.1, this is really a `reminder' about sigma notation which you used in 222 (for series).

• Homework 5, Thur. 9/17/09 : Section 1.6.2: Einstein's summation convention
  • RHOB (Right Handed Orthonormal Basis): sections 1.4, 1.5, 1.6, 1.7. Must understand and be able to explain/derive eqns (14), (15), (16), (21)--(24), (27), (28), (29), (30)--(35).
  • Make your own mental summary of concepts: Geometric meaning and cartesian index expressions for a.b, axb, (axb).c .
  • Summation convention: what is it? Eqns (36)--(39)... what do they all mean? can you rederive them on your own?
  • Exercises 1.6. 1--4.

• Homework 6, Tue. 9/22/09 : Section 1.6.2: Double cross in index notation, Sect. 1.8. Lines, Planes, etc. and other applications
  • 1.6. Write (axb)xc and ax(bxc) in index notation.
  • 1.6. Derive equation (44) from eqn (25).
  • Evaluate the following expression (which use summation convention): (1) ε_ijkε_ijk; (2) ε_ijkε_ijl; (3) ε_ijkδ_ij; (4) ε_ijkδ_kl;
  • Exercises 1.6. All.
  • Exercises 1.8. All.
  • What are the geometric properties (e.g. "it is a sphere of radius 5 centered at the point (2,3,9)") of the set of points whose cartesian coordinates (x,y,z) satisfy
    1. A (x-x₁) = B (y-y₁) = C (z-z₁)?
    2. A (x-x₁) + B (y-y₁) = C (z-z₁)?
    3. (x-x₁)² + (y-y₁)² + (z-z₁)² = A
  • Answer this poor guy's question, and this one too.

• Homework 7, Thur. 9/24/09 : Distance, distance, distance and Rotation, rotation, rotation
  1. What is the distance between the point (x₁,y₁,z₁) and the plane a x + b y + c z + d = 0? Formulate and solve the problem using vectors, specify how to obtain the necessary data from the given data and what actual arithmetic operations must be done to get the actual number. (in other words, the question is not about plugging into a formula, it's about deriving the formula)
  2. What is the distance between the point (4,2,9) and the plane 3x + 2y + z + 3 = 0? (See NOTE above)
  3. How far is the plane a x + b y + c z + d = 0 from the origin? (See NOTE above)
  4. Rotate the vector with cartesian components (4,2,6) about the direction (3,1,5) by angle π/6. What are the components of the rotated vector? (See NOTE above)
  5. Point P has cartesian coordinates (3,2,1). What are its new coordinates after we rotate it by angle π/3 about the axis passing through point A ≡ (1,2,3) with direction D ≡ (4,5,1)? Write the procedure as an "algorithm" (i.e. a `recipe' to follow to obtain the answer from the given data). Write a code (Matlab, C, Fortran,...) to do this for any P, A, D, angle. [ANSWER for this data: new P has coords (0.9494, 3.6934, 0.7351)]. Compare our vector solution procedure to the common matrix solution. Do you feel lucky?
  6. What are the differences between rotating a vector and rotating a point?
  7. Write the vector differential equation dr/dt = ω x r, (i) in index notation, (ii) in matrix notation. Here ω is an arbitrary, constant vector.
  8. If dr/dt = ω x r, with ω constant, show that (i) |r| is a constant, (ii) r . ω is constant, (iii) |ω x r| is a constant. What does that tell you about r? Is anything not constant?!
  9. Find an explicit vector solution for dr/dt = ω x r and verify that it is indeed a solution.
  10. Solve dr/dt = ω x (r-r_A) where ω and r_A are constants. What do these constant vectors represent? explain. Are |r| and r . ω constants? Are there any other constants of motion?

• Homework 8, Tue. 9/29/09 : Sects. 1.9, 1.10: vector function of a scalar variable, e.g. r=r(t) or any a=a(t); Newton's law
  1. Can you prove a product rule? any of them on page 23? How did you prove the product rule in Math 221?
  2. Prove the derivative of a determinant rule using index notation.
  3. What does a . da/dt=0 tell you about a? (i) if this eqn. applies at one specific time, (ii) if it applies for all times.
  4. What does a x da/dt=0 tell you about a?
  5. Suppose you know a and da/dt. Derive expressions for d|a|/dt and d/dt (a/|a|) in terms of a and da/dt (recall that (a/|a| is what we call and write a-hat). Do your equations say what you expect them to say? (i.e. interpret your results)
  6. If v is the velocity of a particle at position r such that v=ω x r, where ω= ω(t), what type of motion is this? What is the acceleration of the particle? What is the acceleration when ω is constant?
  7. If da/dt = ω x a, db/dt = ω x b and dc/dt = &omega x c where ω is not necessarily constant, i.e. ω=ω(t), what is d/dt [det(a, b, c)] = ? Give a geometric interpretation of the result.
     The result is geometrically obvious. The algebraic verification is a bit more involved but not too bad at all, if you know your mixed and double cross product identities. Hint: the Jacobi identity, problem 4, section 1.5 comes in handy.
  8. Section 1.10: Motion with constant acceleration. Rigid body rotation. Motion due to a central force: deduce conservation of angular momentum, Kepler's 2nd law, Kinetic energy and potential energy, conservation of energy.
  9. Explain why r x v = constant, where v=dr/dt, implies that (1) the motion is planar, and (2) `the radius vector sweeps equal areas in equal times'.

• We are skipping sections 1.11 and 1.12. This should be covered in Physics 311.
  You should already know section 1.13.

• Homework 9, Thur. 10/01/09 : Finish up Sect 1.10, begin Sects. 2.1: Change of orthonormal basis. (The online notes do NOT use summation convention but we did use it in class)
  • We solved exercise 1.6 #4 in class. It's an exercise on the summation convention and the `substitution rule' (eqn. (37)). It's 2 lines!
  • Explain why r x v = constant, where v=dr/dt, means that `the radius vector sweeps equal areas in equal times'.
  • Derive conservation of energy and the concepts of kinetic and potential energy from Newton's law for motion of a particle under the action of a central force.
  • I slipped on the last line of lecture which should have read v_i = (e_i . e'_j) v'_j (with summation convention).
  • Deduce the corresponding formula for v'_i in terms of v_j
  • Study section 2.1.

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• EXAM 1 is THURSDAY OCT 8, 11:00-12:15.
  If you signed up for 301 ED SCI (on Johnson), that's where you'll take the exam, otherwise our usual 6102 SOC SCI. This exam is worth up to 25% of your grade!

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• Homework 10, Thur. 10/08/09 : Sections 2.1 and 2.2, orthogonal transformations/change of orthonormal basis. Exercises below use summation convention
  1. Do yourself a favor, think about exam 1 again and try to answer the questions before looking at the EXAM 1 SOLUTIONS!
  2. If (v₁,v₂,v₃) and (v'₁,v'₂,v'₃) are the components of a vector v with respect to two distinct orthonormal bases (i.e. cartesian bases if you wish), how are those components connected? DERIVE the relationships between these two sets of components, both ways v → v' and v' → v. You are not asked to spit back formulas here, but instead to derive and explain what the formulas are and why. You should be able to do this without looking at any notes/book.
  3. If the primed basis is the same as the unprimed basis except that e'₁=-e₁, what is the matrix Q such that v'_i=Q_ijv_j?
  4. If the primed basis corresponds to a rotation of the unprimed basis about the e₂ direction by angle β what is the matrix Q such that v'_i=Q_ijv_j?
  5. If any two sets of cartesian components are related by v'_i=Q_ijv_j? What is the determinant of Q? Does your answer match the two previous specific examples?
  6. If (v₁,v₂,v₃), (v'₁,v'₂,v'₃) and (v''₁,v''₂,v''₃) are the components of a vector v with respect to three distinct orthonormal bases (i.e. cartesian bases if you wish), and v'_i=A_ijv_j, v''_i=B_ijv'_j, (1) what is A_klA_km=?, (2) if v''_i = C_ijv_j, what is C_ij=?, (3) what is C_ijC_ik=?, (4) what is C_ijC_kj=? Can you prove/explain/justify your answers or do we just have to believe you?
  7. The unprimed basis corresponds to cartesian coordinates centered at the center of the earth, with the x direction corresponding to longitude 0, latitude 0 and the z axis as the polar axis. The primed basis corresponds to cartesian coordinates centered at Madison, WI, with the z' axis vertical and the x' axis pointing east. Both systems are right handed. If a vector v has components (v₁,v₂,v₃) in the earth basis, what are its components in the Madison basis? If a point has cartesian coordinates (x₁,x₂,x₃) in the earth basis, what are its coordinates in the Madison basis? Madison, WI is at 43° 4' 23" N, 89° 24' 4" W

• Recommended reading , `Beer and Circus,' sounds familiar?

• Homework 11, Tue. 10/13/09 : On to Vector Calculus , Section 1.1. Curves, parametric representation, arclength, tangent, normal, curvature
  1. Review Chapter 13 in Thomas' Calculus.
  2. Must know and understand parametric equations of lines, segments, circles, ellipses,... (done in class)
     `know and understand' means that you should be able to derive all of this on your own with just a pencil and a sheet of paper.
  3. Must know and understand arclength ds = |dr| ≠ d|r| = dr, unit tangent T = dr/ds, unit normal N such that dT/ds= κ N = N/R, where κ=1/R is the curvature and R is the radius of curvature. (done in class)
  4. Must know and understand relationships between velocity and acceleration and tangent and normal to the curve; curvature and radius of curvature. (done in class)
  5. Given r(t) where t is time, how do you compute the velocity? the speed? the acceleration? the unit tangent? the unit normal to the curve? the curvature? the radius of curvature? (done in class)
  6. Calculate all those quantities for (i) a line, (ii) a circle (done in class)
  7. Consider the binormal B= TxN, where T is the unit tangent, and N is the unit normal. Show that dB/ds is in the N direction.
  8. Vector Calculus notes Study section 1.1, try Exercises 1.2. 1, 2
  9. Given r₀, r₁, ..., r_N, sketch the curve r(t) defined such that r(t)=(1-s) r_n-1 + s r_n for n-1 ≤ t < n, with s = t-n+1 and n=1,..., N.
  10. Quadratic Bézier curve: Given the radius vectors r₀, r₁, r₂, define points A, B and P such that they divide the segments P₀P₁, P₁P₂ and AB, respectively, into equal ratios, that is P₀A/P₀P₁ = P₁B / P₁P₂ = AP / AB ≡ t. Derive the equation that expresses r(t), the radius vector of P, in terms of r₀, r₁, r₂. Make a sketch showing all 6 points.
  11. Cubic Bézier curve: Given r₀, r₁, r₂, r₃, consider the curve defined by r(t) = (1-t)³ r₀ + 3 (1-t)² t r₁ + 3 (1-t) t² r₂ + t³ r₃ with 0 ≤ t ≤ 1. Show that the curve passes through points P₀ and P₃ (easy) but not P₁ and P₂ (less easy) in general (what does that mean?). Calculate the directions of the tangents to the curve at points P₀ and P₃, interpret your results.
  12. Bead on a wire: a bead is launched with initial speed v₀ along a fixed wire. What is the force on the wire if there is no friction and no gravity? Formulate and solve the problem. Make a sketch and clearly define everything you need to formulate the problem.
  13. For current and prospective Physics 311 students mostly: Bead on a rotating hoop: a bead moves along a circular wire that is rotating at uniform angular speed, Ω say, about one of its diameters. (1) Make a 2D sketch that illustrates the problem clearly. (2) Parametrize the position of the bead (i.e. pick a suitable reference point O and a vector basis, and express the position/radius vector r(t) of the particle in terms of suitable parameters). (3) Derive expressions for the bead velocity and acceleration that distinguishes all 3 geometrically important components: tangent to the hoop, perpendicular to the hoop in the hoop plane and perpendicular to the hoop plane. (4) Write F=ma for this problem in the case where the bead is subject to gravity aligned with the axis of rotation and there is no friction between the bead and the hoop. (5) Derive the equations that provide all equilibrium positions of the bead.

• Homework 12, Thur. 10/15/09 : Vector Calculus , Sections 1.1, 1.2: curves and integrals along curves (a.k.a. line integrals).
  1. No! the angle bisector is not a median of a triangle in general, how much other high school math did you miss?
  2. Wait! isn't an integral the ``area under a curve?'' how can we have an integral along a curve? what does that mean?
  3. Study each line of section 1.2.
  4. Exercises 1.2. 4, 5, 6 Visualize the problems! What is dr? what is F? what is F . dr ?
  5. A curve is parametrized by the cartesian x coordinate, i.e. r=r(x) = x e_x + y(x) e_y + z(x) e_z with a ≤ x ≤ b, what is the x-integral for the length of that curve? Derive the formula from first principles, don't just look it up elsewhere.

• Homework 13, Tue. 10/20/09 : Vector Calculus , Sections 1.3, 1.4: Surfaces, surface element, surface line element, surface integrals
  1. Study sections 1.3, 1.4. It may help to review sect. 16.6 in Thomas' Calculus.
  2. What is the meaning of dS, dS, dr, dr = ∂r/∂u du + ∂r/∂v dv ?
  3. Exercises 1.3. 1, 2, 3, 5. #1, #2 done in class, must know how to derive and must know key results.
     In #5, you are expected to deduce the meaning of R and a from the equations that are given.
  4. Exercises 1.4. 1, 2, 3, 4. Visualize the problems! What is dS? what is r . dS ?
  5. (Advanced) Curves on surfaces. Can you prove that the shortest distance between two points on a sphere is the arc of great circle through those 2 points?

• Homework 14, Thur. 10/22/09 : Vector Calculus , Sections 1.5, 1.6, 1.7: parametrizations of volumes. Mappings, change of variables, curvilinear coordinates, Jacobian matrix, Jacobian determinant.
  1. Study sections 1.5, 1.6, 1.7. It may help to review sect. 15.7 in Thomas' Calculus. (except that they use the `American mathematician' notation for spherical coords, not ISO 31-11 . Come to think of it, that `American Mathematician' was probably George Thomas!).
  2. Exercises 1.5. 1, 2, 3, 4
  3. Exercises 1.6. 1, 2, 3, 4
  4. Exercises 1.7. 1, 2, 4, 5, 6

• Homework 15, Thur. 10/29/09 : Vector Calculus , Sections 2.1, 2.2: Rate of change of a scalar field along a curve: f(r(t)), f(r(s)) , Directional derivatives, Gradient, Geometric meaning of gradient, gradient of spherical functions, gradient in cylindrical and spherical coordinates .
  1. Study lecture notes, study problems solved in class and in the notes. There are some Extra notes on GRADIENT .
  2. Consider f(r)= exp(-[(x-x₀)²+(y-y₀)²+(z-z₀)²]/a²), where x₀, y₀, z₀, a are constants. Calculate ∇ f, using both cartesian coordinates and a direct geometric approach. Do you get the same result?
  3. Express ∇ [f(r) g(r)] in terms of ∇ f and ∇ g.
  4. Consider f(r)= x exp(-[x²+y²+z²]/a²), where a is constant. Calculate ∇ f. Express ∇ f in a compact form.
  5. Consider f(r)= z exp(-[x²+y²]/a²), where a is constant. Calculate ∇ f. Express ∇ f in a compact form.
  6. Consider f(r) = r^-1 sin θ cos φ , calculate ∇ f.
  7. A-level: Consider the function f(P)= sum of the distances from point P in 3D Euclidean space to two fixed points P₁ and P₂. Show that the gradient of f at P makes equal angles with the vectors P₁P and P₂P. Sketch. What is the approximate shape of the surface f(P) = constant for very large values of the constant? large compared to what?! What is the smallest value of f(P)? for what P's is that smallest value achieved? What is the approximate shape of the surface f(P) = constant when constant is very close to the smallest value?
     What the heck is this problem about? Solve the problem without using specific coordinates, then pick cartesian coordinates O at the mid point of P₁ and P₂ with z in the P₁P₂ direction, does that choice make sense? What is f(ρ, φ, z) (cylindrical coords)? What is f(x,y,z)? What is the shape of the surface f(P) = constant ? (say, = 2 c) Reduce, practice your algebra, get rid of square roots. What is the geometrical interpretation of this problem?

• Homework 16, Tue. 11/3/09 : Vector Calculus , Sections 2.3, 2.4: Divergence, curl, vector identities
  1. Exercises 2.4. All. Several of these exercises have been done in earlier sections and/or in class.
  2. Swiftly calculate ∇ x (ω x r) where ω is constant and r is the radius vector.

• Homework 17, Thur. 11/5/09 : Vector Calculus , Sections 2.4: exercises, Curl of some basic fields, Section 3.1 (multi-D integrals), 3.2 Fundamental theorem of Calculus
  1. Visualize the vector fields v=v(r)= e_z x r and w=w(r)= v/|v|². Sketch the fields, express them in cartesian coordinates and in cylindrical coordinates. What are physical examples where those fields come up?
  2. Visualize the vector fields v=v(r)= ω x r and w=w(r)= v/|v|², where ω is constant. Sketch the fields, express them in cartesian coordinates for general ω = ω₁ e₁ + ω₂ e₂ + ω₃ e₃. How do these fields compare to those of the previous exercise? What are physical examples where those fields come up?
  3. Solved many exercises from 2.4 in various ways (Gibbs, index, cartesian, cylindrical,...) in class: ∇ . r, ∇ x r, ∇ x (e_z x r), ∇ x B where B is the magnetic field due to a line current B =(e_z x r)/|e_z x r|² (Biot-Savart law for an infinitely long current), ∇ x (ω x r) where ω is constant. Discussed meaning of curl, relation to local rotation, `curl meter' (a.k.a. vorticity meter). Discussed that e_z x r = ρ e_φ, which is easier to visualize.
  4. Calculated ∫_{A → B} ∇ f . dr, for an arbitrary curve (i) `directly' from eqn (52), (ii) through a parametrization r = r(t) and the fundamental theorem of calculus.
  5. Calculate ∫_C ρ^-1e_φ . dr where C is a circle centered on the z-axis, in a plane perpendicular to the z-axis. Show/explain why e_φ = ρ ∇ φ. Pause and reflect. From the previous exercise, shouldn't we find that this integral is zero?! (solved in class 11/10/2009)

• Homework 18, Tue. 11/10/09 : Vector Calculus , Section 3.1 (multi-D integrals), 3.2 Fundamental theorem of Calculus
  1. What's going on in eqns (73) and (74)?
  2. Prove/derive (81) and/or (83) and/or (84).
  3. What's the connection, if any, between (85) and (84)?
  4. 3.7. 1, 2, 3, 4 (with n = ω ).

• Homework 19, Tue. 11/10/09 : Vector Calculus , Section 3.2-3.7: Stokes Theorem, Divergence (Gauss) Theorem
  1. Must know and understand (86). If you're dying for a proof: see page 24. There are other proofs.
  2. Must know and understand (111). Proof is sketched in the notes.
  3. Compare (86) and (111), note the differences. You'll be expected to know those theorems and how to use them.
  4. Proof of (111) actually involves (113) and (113) is more flexible. You should be able to derive (115)-(118) from it.
  5. What is the meaning of curl? see and ponder (87).
  6. What is the meaning of divergence? See and ponder (112)
  7. What is the meaning of gradient? Hey! you should know that already.
  8. 3.7. 1, 2, 3, 4 (with n = ω ) from HWK 18 and now also 5 -- 11. (12 not assigned but important classic, cannot understand/solve 12 if you did not understand 9 then 10, then 11 first.)
  9. The fields E=r/r³ and B=(e_z x r)/|e_z x r|² = ρ^-1e_φ, what are their divergence, curl, circulations about arbitrary closed curves, flux through arbitrary closed surfaces? What do these fields looks like, can you visualize them?

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• Homework 20, Thur. 11/19/09 :
  • Do yourself a favor, think about exam 2 again and try to answer the questions before looking at the EXAM 2 SOLUTIONS!

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• Homework 21, Tue. 11/24/09 : Functions of one complex variable Section 1.1-1.8: Complex numbers, geometric series, Taylor series, exponential, log, roots,...
  1. Math 222 material review and polish:
     • What is a complex number? how to add, multiply -- and divide complex numbers
     • What is the real part of i ? What is the imaginary part of i ?
     • What is a `complex conjugate'?
     • What is the `modulus' of a complex number?
     • What is the `argument' of a complex number?
     • What are all the z's such that |z|=1?
  2. 1.1. 1, 2, 3
     • What is a geometric sum? How do you derive the compact formula for the geometric sum?
     • What is a geometric series? For what complex numbers does it converge? diverge?
     • What happens when |q|=1? Try it out for a few different such q's
     • If you borrow A dollars at annual interest rate r, compounded monthly for a term of N years, what are your monthly payments? Explain how to determine the monthly payments, that is: derive the formula. What does this have to do with this section?
  3. 1.3. 1, 2. Prove formula (8) using geometric sums.
     • What is a useful convergence test for series? Why does that test work?
     • What is a power series?
     • Where in the complex plane does a power series converge? where does it diverge?
     • What is a Taylor series?
  4. 1.5. 1--7. (You know and understand formula (17), don't you?)
     • What are the Taylor series of exp(z), cos(z) and sin(z)? You know those, don't you?
     • What is the `Proof that God exists?'
  5. 1.6. 1, 2, 4, 5, 6. (You know formula (22), (23), (24), (25), (26), (31), (32) and (33), don't you? If not, you really should!. You saw all these for a real variable x, and they apply likewise for a complex variable z).
     • What is the polar form of a complex number? for what operations is the polar form particularly convenient?
  6. 1.7. You understand that formula (39), (40) and (41) directly follow from Euler's formula (33) so you know them, don't you?
     • What is the log of a complex number? What is ln(i)? What is ln(-1)?
     • What is a complex exponential? What is iⁱ?
  7. 1.8. 1, 2, 3(i) and (ii). (if you've got the energy, look at (iii) it's cool). You know and understand formula (46), (47) and (50), don't you?
     • What are the `roots of unity'?
     • What is `factoring a polynomial'? Can you always do that?

• Homework 22, Thur. 12/03/09 : Functions of one complex variable Section 2: complex differentiation, Cauchy-Riemann, orthogonal coordinates, conformal mapping.
  1. What are the Cauchy-Riemann equations? where do they come from? What do they mean geometrically? What is the relation with Laplace's equation?
  2. Is f(z)=|z| differentiable? Explain/prove. Can you explain/prove without doing any calculations? (hint: what is a complex derivative? where is |z| constant?)
  3. Is f(z)=z* differentiable with respect to z?
  4. If we consider the mapping z → w=f(z) from the complex z=x+iy plane to the complex w=u+iv plane, when are angles preserved? What does it mean that angles are preserved? How do you show/prove that angles are preserved? Where are angles not preserved?
  5. Exercises 2.2.1, 2, 3. You might want to look at the complex map examples, after having tried to figure out the problems for yourself, of course!

• Homework 23, Tue. 12/08/09 : Functions of one complex variable Section 3: complex integration
  1. Must know and understand (i.e. know how to derive/justify): (1) Meaning of an integral in complex plane, (2) how to explictly compute an integral using a curve parametrization (e.g. 2 examples on page 15), (3) Cauchy's Theorem, (4) Justification of `contour deformation' (picture and argument on page 16), (5) application to integral of (z-a)ⁿ, n=0,+/- 1, +/- 2, ... , (6) Cauchy's formula and its derivation (pg. 18), (7) Cauchy's general formula and at least a quick and dirty justification/derivation (eqns. (74), (79)),
     Advanced theory (for A+, but no one gets an A+ officially...): Derivation and justification in eqns (76)--(80), deep results about function of a complex variable: differentiable in a domain => infinitely differentiable (holomorphic) AND Taylor series converges in some disk (analytic).
  2. Exercises: 3.1. 1, 2, 3, 4, 5, 6, 8, 9.
  3. Exercises: 3.2. 2, 7

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• EXAM 3, THURSDAY 12/10/09, 5:00-6:15pm in VV B239 ALL Material on complex variables, `elementary' complex functions (including exponential, sine, cosine, log, complex exponentials and roots), and functions of one complex variable, EXCEPT section 4 of the notes. Section 4 WILL be on the final next THURSDAY DEC 17.

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• Homework 24, Mon. 12/14/09 : Functions of one complex variable Section 4: Real applications of complex integration
  1. Study all solved examples in section 4. Understand procedure and how to justify that some contour integrals go to zero in some limits.
  2. Exercises 4.1, 2, 3, 5