PART 1: Vectors.

• Homework 1, Wed 1/23/2008 : STUDY SECTIONS 1.1, 1.2. Note that we will be primarily concerned with "prototypical vectors" i.e. displacements in 3D (Euclidean) space in this class, the vector concept originates from those displacements in 3D space but is much more general.
  • 1.1.4
  • 1.2. 1 (in (i) BOTH &alpha and &beta are in [0,1]), + problems 5--8
• Homework 2, Fri 1/25/08 : STUDY SECTIONS 1.1, 1.2, 1.3.
  • Digest the diagrams accompanying formulas (12) and (13). You want to understand the geometry and the concepts and be able to deduce the formulas (12) and (13) from those concepts. These are very basic applications of the dot product that come back in many other problems and applications.
  • Deduce the 5 properties of the dot product (top of page 7) from its geometric definition.
  • 1.3. 1--8, do 1.3.4 two ways: geometrically and vector-algebraically
• Homework 3, Mon 1/28/08 : STUDY SECTIONS 1.1, 1.2, 1.3, 1.5. Note that we are covering the material in an order slightly different from that in the notes. We're getting the geometric concepts down, then will discuss orthonormal bases and index notation.
  • Get comfortable with the right hand rule (a.k.a. corkscrew rule) and the geometric definition of the cross product.
  • Prove the distributivity property (3, p. 11) geometrically.
  • 1.5. 1--3, 5. (1.5.3 geometrically, as done in class, for now).
• Homework 4, Wed 1/30/08 : STUDY SECTIONS 1.1, 1.2, 1.3, 1.5, 1.7 (geometrical aspects for now, we have not yet discussed orthonormal bases and index notation).
  • You must known and understand the double cross product, formula (25), know how to remember it and why it is the way it is from the geometric definition of cross product.
  • 1.5. 3 (now using the double cross formula), 4.
  • Mixed product, formula (46). Know and understand the geometric interpretation as signed volume , give a geometric argument for the identity (a x b) . c = (b x c) . a = (c x a) . b. Read and understand page 18 and 19 (up to formula (53) for now).
  • 1.7. 2, 3, 4, 7 (using vector identities for now), 8, 9, 10 (hint: 8, 9, 10 are directly related).
• Homework 5, Fri 2/1/08 : STUDY SECTIONS 1.4 (Orthonormal bases, Kronecker delta &delta_ij, dot product for orthonormal bases, sigma notation), 1.6 (Levi-Civita symbol, &epsilon_ijk: definition and connection with right-handed orthonormal bases.)
  • 1.4. 1--5 (get comfortable with &delta_ij and sigma notation).
  • Sect 1.5. read `orientation of bases', derive formula (24).
  • Sect 1.6.1 read, next lecture will be 1.6.2 Einstein's summation convention
• Homework 6, Mon 2/4/08 : STUDY SECTION 1.6 (Index notation, Einstein summation convention) It is very important to get this under control NOW! this is a widely used convenient notation that we will use throughout this course and beyond. Study section 1.6 carefully.
  • 1.6. 1--4
  • 1.7. 11, 12, 13 (not index notation, but these exercises were not assigned earlier).
• Homework 7, Wed 2/6/08 : STUDY SECTION 1.6, 1.7. You must become comfortable with index notation, summation convention, formula (44) catch up with all previously assigned exercises.
  Next lecture: section 1.8.
• Homework 8, Fri 2/8/08 : Review and clean-up your Quiz 2 , STUDY SECTION 1.8 You must become comfortable with the coordinate-free vector equations for lines, planes etc. You must understand implicit equations and parametric equations, you must be able to go from coordinate-free vectors to cartesian representations and vice-versa.
  • Solve all problems in sections 1.8: Distance from a point to a plane, a point to a line, between 2 lines, etc.
  • For all the following problems: solve in coordinate-free form first, then translate into cartesian coords:
  • What is the distance between the point (x<sub>P</sub> ,y<sub>P</sub> ,z<sub>P</sub>) and the plane A (x-x<sub>1</sub>) + B (y-y<sub>1</sub>) + C (z-z<sub>1</sub>) = 0 ?
  • What is the distance between the point (x<sub>P</sub> ,y<sub>P</sub> ,z<sub>P</sub>) and the plane A x + B y + C z= D ?
  • What is the distance between the point (x<sub>P</sub> ,y<sub>P</sub> ,z<sub>P</sub>) and the line (x-x<sub>1</sub>)/A= (y-y<sub>1</sub>)/B= (z-z<sub>1</sub>)/C ?
  • What is the distance between the lines A<sub>1</sub> (x-x<sub>1</sub>)= B<sub>1</sub> (y-y<sub>1</sub>) = C<sub>1</sub> (z-z<sub>1</sub>) and A<sub>2</sub> (x-x<sub>2</sub>)= B<sub>2</sub> (y-y<sub>2</sub>) = C<sub>2</sub> (z-z<sub>2</sub>) ?
• Homework 9, Mon 2/11/08 : Sect. 1.9. and all exercises solved in class (derivative of the norm of a vector, geometric understanding of the result, derivative of a vector of constant magnitude, ... )
• Homework 10, Wed 2/13/08 : Sect. 1.10: Motion under a central force: conservation of angular momentum, planar motion, Kepler's law, conservation of energy. Do you understand all those things, can you derive them from Newton's law? Do these only work for `inverse square' forces (like gravity and Coulomb forces)?
  • A particle is at position r(t) at time t. It is at its perigee (with respect to O) at time t_p and at its apogee at time t_a. What is the mathematical definition of perigee and apogee? What can you say about the velocities v(t_p) and v(t_a)? [NOTE: nothing is said about the particle moving according to a central force, so don't assume that!] Now, if the particle IS moving due to a central force always pointing toward O, what more can you say about v(t_p) and v(t_a)? Perigee, apogee? what the heck are those?!.
• Homework 11, Fri 2/15/08 : Sect. 1.10: Rotation! There is not much in the notes but we did a lot more in class. There are extra notes that cover that material Solution of dv/dt = b x v solution (motion of charged particle in a constant magnetic field; uniform rotation) This is essentially identical to simple rotation at &omega.
  1. How long did it take you to read and digest the Solution of dv/dt = b x v? Do you feel confident you digested them? What quiz or exam question do you expect on those notes? What is unclear to you?
  2. Write dr/dt =w x (r - r_a) in cartesian coordinates. If you took Math 319 or 320, can you and how would you solve the resulting system? (with w and r_a constant). All of you should be able to solve this using vectors and interpret the solution geometrically.
  3. Same question but for du/dt = c x u + d, with c, d constants. Write in cartesian, solve using math 319/320 methods if you can. Solve using vectors. Describe what happens to u as a function of time t. [Hint? how about thinking parallel to c and perpendicular to c?]
  4. OK, in class, we constructed the equation for general rotation about an arbitrary axis starting from what we know for simple rotation about the z axis in the xy plane. Now suppose you are given the equation du/dt =c x u where c is constant but you don't know the solution (that's usually what happens with equations, we have to solve them). Analyze this equation to deduce key properties of its solution. What does the cross product beg you to deduce? Don't just say "the solution is (...) because the prof told me it was and I memorized it by heart." No, use the equation to deduce key properties.
  5. An object is described by 7648 points with cartesian coordinates (x_n,y_n,z_n), n=1,...,7648, e.g. the first point is (x₁,y₁,z₁)=(4,2,7). Those 7648 points are a crude finite element model of a new Nuclear reactor. I spare you the other 7647 triplets of coordinates. You want to rotate all those points by an angle &alpha about the line through the point (3,9,1) and parallel to the vector (5,8,2). Mama mia! How do you do that? relax, you've got fast computers now, but what do you tell the computer to do?
• Homework 12, Mon 2/18/08 : Sect. 1.10: Rotation! Continued Did you do all the previous homework? We essentially solved problems 1, 5 in class. Where does that point (4,2,7) end up after that rotation by &alpha in #5?

PART 2: Vector Calculus.

• Homework 13, Wed 2/20/08 : READ SECTIONS 1.1, 1.2
  • 1.2. 5, 6.
• Homework 14, Fri 2/22/08 : STUDY SECTIONS 1.1, 1.2. You must understand the meaning of r(t) and its derivative. You must understand the meaning of the line element dr and of the various line integrals. "Line integrals" are integrals along curves (so curve or path integrals would have been better nomenclature) but they usually have nothing to do with "area under the curve" that you saw in elementary calculus of one variable! Don't be afraid to look back at your elementary calculus book where you discussed parametrization of curves and line integrals. (Math 222 and 234) If you used Varberg 8th edition, that's in Chaps 12-14: Polar coords, ellipse etc.: Varberg Chap 12, Curves: Chapter 13 and sections 14.4, 14.5, look at them, it's all there!!! Unfortunately they use those old i, j, k unit vectors and they only box the cartesian formulas.
  • 1.2. 1--6.
  • We deduced in class that the length L of the ellipse of major radius a and minor radius b is bigger than 2 &pi b and less than 2 &pi a, that is: 2&pi b < L < 2 &pi a (less or equal really since the circles are extreme cases of the ellipse). That seemed intuitive from the picture we drew to elucidate the geometric meaning of the angle &theta . But is it really true? Prove it ! (using parametric representation and definition of length of a curve as a certain "line" integral.)
    Is it possible to have a curve lying in between the concentric circles of radii b and a whose length is less than 2 &pi b? larger than 2 &pi a ?
  • 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" ?
    • 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 15, Wed 2/27/08, Fri 2/29/08 : STUDY SECTIONS 1.3, 1.4. You must understand the meaning of r(u,v), its partial derivatives and the cross product. You must understand the meaning of the surface element dS. "Understanding" means, among other things, being able to use the concept in various distinct explicit calculations, not just "Oh, yeah, I understand". If you used Varberg 8th edition, you may want to look back at Sections 14.6 (surfaces in 3-space), 16.1-3 (double integrals), 16.6 (surface area), 17.5 (surface integrals) , but you certainly want to study sections 1.3, 1.4 of our vector calc notes.
  • When are coordinates for a curved surface called "orthogonal"? Does this concept make sense to you? What is |dS(u,v)| when (u,v) are orthogonal coordinates for the surface?
  • What is the integral of dS over the surface of a sphere of radius R? Over the surface of a donut? of a submarine? of a tetrahedron ?
  • What is the integral of |dS| over the surface of a sphere of radius R? of a tetrahedron ?
  • 1.3. 1--5 (warning, some of these involve a fair amount of calculations. You must do all the calculations for 1.3.1 and 1.3.2. Problem 1.3.4 involves the 3 Euler angles as in problem 1.2.3, you do NOT need to do all the calculations but be realistically confident that you could do them, i.e. what would you need to do to obtain f(x,y,z)=0, g(x,y,z)=0, the equations of the ellipse in terms of (x,y,z), what general features do these equations have? Hint: x₁ = r . e₁ where r = x e_x + y e_y + z e_z .
  • 1.4. 1--4 (spherical coords are in Varberg 8th, sect. 14.7: but they use &rho as distance to the origin, &theta as longitude and &phi as polar angle. Applied Mathematicians, Engineers and Physicists (AMEPs) tend to use r as distance to O, &theta as polar angle. AMEPs use &rho as distance to the z-axis, as we have done in class.)
  • (Just for the fun of it) Find a parametrization for a sphere in 4-space: x₁² + x₂² + x₃² +x₄² = R² . Why the heck would anyone worry about a sphere in 4-space you ask? Well, think of two mass spring systems connected to each other: the total energy of this system (kinetic + potential) is fixed. That system lives in 4-dimensional state space: 2 positions (one for each mass) and 2 velocities. Conservation of energy in some suitable variables means that the point in 4-dimensional state space representing an instantaneous state of the system is moving on a sphere in 4-space.
• Homework 16, Mon 3/10/08 : Study sections 1.6, 1.7.
  • 1.6. 1--4
  • 1.7. 1--7
• Homework 17, Wed 3/12/08 : Study sections 2.1, 2.2. Look back at your calculus book for examples and illustrations of isocurves, isosurfaces and gradient in addition to what we did in class. You must understand gradient geometrically, not just know the x,y,z formulas.
• Homework 18, Fri 3/14/08 (posted Mon 3/17/08) : We took another look at the gradient and derived its expression in spherical coordinatess . Study these notes and do the exercises given there. Next: divergence, curl and vector identities. Review index notation and &epsilon_ijk , you're going to need it!!!
• Homework 19, Wed 4/2/08 (cumulative) : Sections 2.3, 2.4: Grad, Div, Curl: make sure you understand/know those and do not confuse them. You must be able to rederive all vector identities in 2.4 using Gibbs notation and to prove them using index notation.
  • 2.4. All, many are repeats of problems done earlier or have been done in class. For #9, you must be able to do it using cartesian coordinates as well as using vector identities (as done almost completely in class today). [We used vec identities to show that &nabla x (e_z x r) is too easy , I mean, 2 e_z. Explain/show why | e_z x r | = &rho = distance to z-axis. Then why &nabla &rho^-2 = -2 &rho^-3 e_&rho (what is e_&rho ?). Then why e_&rho x (e_z x r) = &rho e_z. Put it all together using &nabla x (f v) identity to show that &nabla x [&rho^-2 (e_z x r) ] = 0, as long as &rho is not 0 (i.e. away from z-axis where Mag field is singular).
  • 3.1--4 You must be able to prove (81) or (83) or (84). You must know and understand (86). Just knowing it is not enough, you must understand what S and C are and how they are connected. The proof of (86) falls back on (84) using index notation. That's on page 24, it's an excellent exercise.
  • 3.7. 1--4. #1 were done earlier in one way (what way? see eqn (52) in 2.1 and gradient exercises), remind yourself of that and compute them in a new way also, using the new theorems covered in these sections. Same for #2 which is basically a special case of 1 (ii). #3 is "tricky," but not in a gratuitous math puzzle way, it's a fundamental physical problem, it's "Ampère's law" for a line current. Parts of it were done in class. #4 was basically done in class.
• Homework 20, Mon 4/7/08: Section 3: The various versions of Fundamental Theorem of Calculus: organize them in your mind, make sure you understand what's involved: what kind of function, what kind of curve, surface, volume? Dont' confuse Stokes and Gauss theorems. They are broadly similar but it is common for students to get confuse and mixe them up incorrectly.
  • Be able to explain the proof of result (107).
  • Be able to deduce (111) from (110) and vice versa .
  • Understand/be able to explain that (107) can be written in the more general form (113).
  • Understand/be able to derive all the various forms of the "divergence theorem" (section 3.7). In particular, (111) [divergence theorem] , (113) [general form of fundamental theorem] , (114, 115) ["gradient theorem"] , (117, 118) ["curl theorem"] [NOT TO BE CONFUSED WITH STOKES!!]
  • 3.7. 5--11. Note that 3.7.12 is a relatively easy extension of 3.7.9-11. This is a very important result in physics known as "Gauss' law" or Gauss' law for gravity .

PART 3: Complex Calculus .

• Homework 21, Mon 4/14/08: Sections 1.1, 1.2, 1.3: Complex algebra, geometric series.
  • 1.1. 1, 2 + Must know basic complex algebra concepts and definitions inside out, in particular how to divide complex numbers, and binomial formula.
  • 1.2. 1, 2 + Must understand concepts and definitions of limit, continuity and derivative.
  • 1.3. 1, 2 + Read the tiny section 1.3 very carefully.
• Homework 22, Wed 4/16/08:
  Section 1.4: Series, Ratio Test, power series, Radius of convergence, Taylor series.
  Section 1.5: Definitions of exponential, cosine and sine for complex variable z
  • Must digest paragraph following eqn (14).
  • Must know Taylor series formula (15).
  • Must understand/know how to derive eqns (16)--(19)
  • 1.4. 1, 2, 3, 4 (done in class), 5, 6, 7
  • Must know eqns (20), (21), (22), (28), (29), (30).
  • 1.5. Elementary (pre-321): 1, 2, 3, 4. Current (321-level): 5, 6, 7.
• Homework 23, Mon 4/21/08: Section 2: Cauchy-Riemann, contour plots.
  • 2.1 1, 2, 3
  • 2.2. 1, 2 (see some solved exercises posted on top page )
  • Matlab complex demos
• Homework 24, Wed 4/23/08: Section 2: Conformal mappings.
  • 2.2. 1-4
• Homework 25, Mon 4/28/08: Section 3: complex integration: what is a complex integral? Cauchy's theorem?
  • 3.1 Must know/understand proof of Cauchy's theorem
  • 3.1 Must know/understand eqn. (64)
  • 3.1 1--10
• Homework 26, Wed 4/30/08: Section 3.2: Cauchy's formula, mean value theorem, generalized Cauchy formula, analytic function.
  • 3.2 Where does formula (66) come from? can you derive it? What is &zeta in (68)?
  • 3.2 Can you derive (71) from (66)?
  • 3.2 Can you explain what's going on in (73) to your mechanical engineering friend who should have taken Math 321 but didn't?
  • 3.2 If you want to understand how to do (73) cleanly, read and understand the paragraph between (73) and (74). There we see what needs to be done to make sure that all the limits can be interchanged. Good for you, and once again it involves geometric series.
  • 3.2 Read, understand and know what's going on in (75), (76), (77).
  • 3.2 All: 1--6
• Homework 27, Fri 5/2/08: Section 3.3: Applications of Complex integration.
  • 3.3 The method in eqn (79) should be a piece of cake to you. We did a more general version in class today, together with a quick discussion of Fourier series as background motivation for these types of integrals. What we did in class is equivalent to integral (80).
  • 3.3 1.
• Homework 28, Mon 5/5/08: Section 3.3: Applications of Complex integration continued.
  • 3.3 (81) and (82) done in class. Try to redo it on your own without looking at your notes. Refine your solutions, it's very easy to spend a lot of time doing unnecessary algebra. Not good on an exam.
  • 3.3 Prove that the integral of 1/(1+z⁴) over the semi-circle C₂ : z = R e ^{i &theta} for &theta= 0 to &pi, goes to zero as R goes to infinity.
  • 3.3 compute the integral of 1/(1+x²+x⁴) over the real line. Remember to prove that the integral over "C₂" goes to 0.
  • 3.3 compute the integral of cos(k x) /(1+x²) over the real line, where k is a real constant. Why can you assume that k > 0 (or k < 0 )? Don't forget to prove that the integral over "C₂" goes to 0!! That's a big part of this problem.
  • 3.3 try out (83) and (84) especially. ASK in lecture on Wed 5/7/08 if you have questions, or forever hold your peace.
  • 3.3 2.
• Homework 29, Wed 5/7/08: Section 3.3: Applications of Complex integration continued.
  • 3.3 Discussed the famous and tricky integral (87), together with Jordan's Lemma , (91), a nice little piece of analysis. Sketched quickly the idea for Fresnel integrals which arise in Fresnel diffraction and other dispersive wave problems.