DUE Fri Sept 12: Hand-in ONE sheet (A3 format=8.5in x 11in) with the clean, readable solutions of problem 1.4.3 and 1.4.8 (one on each side ideally). Make sure to have all your dots and underlying wiggles where they should be. The solutions will be read by a computer that cannot fill-in missing bits and pieces for you. Write your name AND YOUR SECRET 4 LETTER CODE on the sheet.

DUE Mon Sept 22: #6 below in the list of 1.1.1.-1.1.4 extras, 1.8.5 (Laplace's eqn), 1.9.2 (gravitational field due to a spherical shell. Newton had trouble with this one, what about you?), area of arctic circle (see below) NAME and (same as last time) 4 letter code again. Up to 10% of grade will depend on presentation! Write-up must be C³: Clean, Clear and as Concise as possible (without hampering clarity. This is a constrained optimization process).

EXAM 2 is FRI NOV 14 !!!!

Practice for your own safety :

Chap 1

• 1.1-1.4 Vectors: dot, cross, double cross, mixed products and applications
• 1.1. 1, 2, 3, 4, 6, 9
• 1.2. 1, 2, 5
• 1.3. 1, 2, 4, 6, 8, 9, 11, 13, 14
• 1.4. 1, 3, 4, 5, 6, 8, 14
• 1.1-1.4 extras:
  1. Assume that the vectors a, b, c are co-planar. Write c as a linear combination of a and b (obtain a general formula).
  2. Find a vector x that is perpendicular to a and such that the cross-product a X x= a X b (general formula).
  3. Obtain explicit formulas for (1) and (2) for a=(1,2,0), b=(2,1,0), c=(4,3,0).
  4. Given two vectors a, b, find all the vectors x s.t. a X x = b.
  5. Same question, but for vector eqn a X x = b-x
  6. The basic force balance for the large scale motions in the atmosphere is Coriolis = pressure gradient, or 2 W X v = - grad P, where W is a constant rotation vector orthogonal to the earth surface, v is the wind velocity vector and P is the kinematic pressure. Solve for the wind velocity. Think of the weather maps showing regions of High and Low pressure. What is the wind direction between a High and a Low in the northern hemisphere (where W points "up")? What is the wind direction on an iso-contour of pressure (= isobar)? around a High? around a Low? Make sure a skeptic who understand vector calculus would believe you.
• 1.5-1.8 Vector Analysis: grad, div, curl and all that
• 1.5. 2, 4
• 1.6. 1, 2
• 1.7. 1, 2, 3, 7-11, 15
• 1.8. 2, 3, 5
  
• 1.9-1.11 Vector integration: Line and surface integrals. Volume integrals. Gauss and Stokes theorems.
• 9. 1, 2, 3. P 64 Correction: If an area is defined implicitly by the equation F(x,y,z)=0, then its unit normal is n = grad F/||grad F|| (correct? or do you believe everything you read?). For F(x,y,z) = z-f(x,y)=0, grad F=(-df/dx,-df/dy,1) (where the derivatives are partial derivatives).
  1. Calculate the area of Earth that lies north of the arctic circle. Yes, you need to figure out what that means. You can find what you need to define the arctic circle here and the basic earth data required at this site. But work with appropriate, descriptive letters, THEN substitute the numerical values at the very end.
  2. Calculate the integral of y dx + x dy along the parabola y=x² from (0,0) to (2,4).
  3. Calculate the integral of y dx - x dy along the circle of radius 1.
  4. Calculate explicitly the work done by the gravitational force m g (constant) on a particle of mass m moving between points A and B. Show by explicit calculation that this work is independent of the path.
• 1.10. 1, 2
• 1.11. 1
• 1.12. 2
• 1.13. 2

Chap 2

• Polar coordinates 2.2. 1, 2, 3, 5, 6 (read and understand example 2.2.5 first), 7, 8, 9, 10
  Planetary motion:
  1. Complete the derivation of the orbit equation for rho(phi) , sketched in class. Sketch the orbits for epsilon greater than 1, =1, less than 1 and =0. Kepler's 1st Law stated that the orbits of planets and comets are ellipses for which the Sun is one focus.
  2. Write the orbit equation in Cartesian coordinates. Show that the half major axis a= p₀/(1-epsilon²) , and the half minor axis is b= p₀/sqrt(1-epsilon²) where p₀ = L₀²/ k was defined in class.
  3. In the case of an elliptical orbit: show that the area of the orbit A= pi a b = L₀ T/2 where L₀ is the constant angular momentum and T is the period of rotation around the ellipse. Show that T ² =4 pi² a³/k and deduce Kepler's 3rd Law that The square of the period of rotation of planets and comets is proportional to the cube of the major axis .
  4. Show that if you know a, b and T for one planet (i.e. shortest and longest distance to the Sun, and period) then you can determine the gravitational constant k = G M_sun which is the same for all planets and comets.
  5. Assume that the Earth orbit is a circle of radius 149,600,000 km and deduce k. Now solve problems 2.2.6 and 2.2.7.
  6. The two-body problem: We "neglected" the motion of the Sun but show that if two masses m₁, m₂ at position r₁ and r₂, respectively, exert on each other a force along the direction r₁-r₂ whose magnitude depends only on the distance between them (and satisfies the "action-reaction" principle) then the motion of the two masses reduces to studying the motion of the single mass m = m₁ m₂/ ( m₁ + m₂) at position r=r₁-r₂ under the action of a central force. Show that if m₁ is much bigger than m₂ then m is approximately equal to m₂ , the smaller of the two masses.
• General coords 2.3. 2, 3, 4. (For fun try the more advanced: 5, 6).
  Extras:
  1. Let x = cosh u cos v and y = sinh u sin v. Sketch the u = Constant and v = Constant curves. Determine (1) the unit vectors of the u and v coordinates and the length (``metrics'') h_u, h_v; (2) the vector line element dr and the arclength ds; (3) the surface element in u, v coordinates. Calculate the area between the curves defined by u=u₀ and u=u₁.
  2. For a perfect gas the pressure p, volume V and absolute temperature T are related by the equation: pV= R T, where R is a constant. An isothermal transformation corresponds to T= constant, and therefore pV=constant (not the same constants!). An isentropic transformation on the other hand, corresponds to pV ^g= constant, where g is a constant corresponding to the ratio of specific heats g=C_p/C_V. Consider a Carnot engine in which a perfect gas goes through a cyclic transformation consisting of an isothermal expansion (pV=u₂), an adiabiatic expansion (pV ^g= w₂), an isothermal compression (pV=u₁ < u₂) and an adiabatic compression (pV ^g= w₁ < w₂). The mechanical work performed by the gas is equal to the area between the curves pV=u₁, pV=u₂ and pV ^g= w₁, pV ^g= w₂, in the p V plane. Calculate that area. [Hint: do a certain p, V ---> u, w change of variables and use the fact that det(dx_i/dq_j) = 1/det(dq_i/dx_j) (i.e. the Jacobians are inverses of each other, see curvilinear coords notes) where the two sets of coordinates are here (p, V) and (u,w).].
  3. Calculate the volume and area of a torus (or "donut" if you prefer).
• Differential Vector Operators 2.4. 1, 2 (using divergence and Stokes Theorem, as in section 2.4), 3
• Spherical Coords 2.5. 1, 2, 3, 6, 7, 8,
• Rotation of Coordinate axes See relevant parts of sections 2.6 and 3.3.
  1. Find the rotation matrix (i.e. the table of "direction cosines" a_ij) for a right-hand rotation by an angle Phi about the z-axis.
  2. Same question for a right-hand rotation by an angle Alpha about the y-axis.
  3. Can you find the rotation matrices for right hand rotation by an angle Phi about the axial directions (1,1,0) and (1,1,1)? [Hints: tensors are the easy way to deal with this, but that's for a bit later. For now, think that Phi rotation would be easy if the axial direction was the "z" direction (especially if you just figured out 1 above). It's not the z-direction but we can think of successive rotations bringing the original z into the desired z directions. Thus we can rotate the z axis into the x-axis by a pi/2 right hand rotation about the y-axis. Then we can rotate that new z', say, by a -pi/4 right hand rotation about the new x' axis. The new coordinates are x'', y'', z''. They are related to the original coordinates by 2 successive rotation matrices. Now all we need is a right hand rotation by Phi about that new z''. You can try to figure it out rightaway but it's more general and systematic to break it up into successive rotations and intermediate coordinates (x,y,z) -> (x',y',z') -> (x'',y'',z'') -> (x''',y''',z'''), as just sketched for the (1,1,0) case].
• Levi-Civita (or "alternating") Symbol 2.9.3, 4, 5, 7, 8, 9, 11
• Tensors See exercises on handout

Chap 3

• Extra exercises on determinants (4 page handout distributed on 10/29/2003) +
• 3.1. 2, 3, 4, 5* (not easy), 7. Solve the system of three equations x + 2 y + 3 z=1, 2x + 4 y + 5 z =1, x + 2y + 2z=0 using Gaussian Elimination> You must indicate and perform your steps clearly (for example "Eqn 2 := Eqn2 - 73/4 Eqn 1, Swap Eqn 2 and Eqn 3,...).
• 3.2. 1, 4, 5, 8, 19, 24, 26.
• 3.3. 1, 2, 3, 12

Chap 5 Series fundamentals (Calc 222 review of some choice morsels)

• Read Example 5.1.2 and section 5.2, in particular the "ratio test". Try exercise 5.1.1.
• You're supposed to know section 5.6: Taylor series. You should know Taylor's formula and how to use it, the Taylor series of e ^x, sin(x), cos(x), ln(1+x),... and be able to show that those series converge. You should also known the binomial coefficients and the expansion of (1+x)ⁿ for any positive integer n. (The binomial theorem extends this expansion to general n).

Chap 6 Functions of one Complex Variable READ ALL TEXTBOOK SECTIONS CAREFULLY!

• 6.1. 1, 4-9, 12, 14, 22 [This means: READ 6.1. THEN TRY exercises 1, 4-9, 12, 14, 22, then READ AGAIN 6.1. if needed, THEN TRY AGAIN 1, 4-9, 12, 14, 22, etc... This is usually a convergent process, in fact it often converges in a finite number of iterations!!] NOTE: the ANS. to 6.1.22 is not quite right, what is the correct answer?
• 6.2. 1, 2, 5, 7 [i.e. READ 6.2. THEN TRY exercises 1, 2, 5, 7 then READ AGAIN,...]
  Extras:
  1. consider f(z)=z² = u(x,y) + i v(x,y) , sketch some of the curves u(x,y)=const. and v(x,y)=const. ,
  2. Same question for f(z) = e ^z . Also, sketch the mapping w = e ^z , i.e. show where characteristic sets (such as horizontal and vertical lines, radials, circles,...) in the z-plane are mapped in the w-plane.
  3. Consider a function f(z) of the complex variable z, let f(z) = u(x,y) + i v(x,y) where u(x,y) and v(x,y) are real, by definition. Show that if u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations then df/dz (defined as the usual limit) does not depend on how that limit is taken.
  4. Calculate df/dz for f(z)=z^* (i.e. f(z)= conjugate(z)).
  5. What is an "analytic" function? what is a "holomorphic" function? What is an "entire" function?
• 6.3 1, 3, 4, 5 (and of course READ the section and make sure you digest all examples very well).
  Extras:
  1. Complete the integral of 1/z over a square as started in class. The eqn and procedure following eqn (6.36) in the book is not correct (why?). If you think before computing (usually a good thing to do) there is no integral to compute!
  2. Compute the integrals of z² and e^z over the path (a) 0 -> 1 -> 1+i, (b) the circular arc from 4 to 4i, (c) the circle of radius 3 centered at 4+4i.
  3. Prove Cauchy's theorem using the divergence (Gauss) theorem.
• 6.4 1, 2, 3, 8, 9
  Extras
  1. Complete in full details the calculation of the integral of 1/(1+x²) over the real axis using complex integration, as done in-class for the most part.
  2. Calculate the integral of e^z/(z(z²+9)) over the circle of center z=1+i and radius 2 by (1) the "cut-out method" and explicit parametrization of small circles, and (2) by Cauchy's integral formula.
  3. Calculate the integral of z² from z=1 to z=1+i. Discuss.
  4. Calculate the integral of z^(-1) from z=1 to z=1+i. Discuss!
  5. Calculate the integrals of cos(z)/z and sin(z)/z over a circle of radius 2 centered at the origin.
• 6.5 1, 2, 8, 13. Pretty good section to read, but not great exercises, so here's some
  Extras:
  1. What is the Taylor series expansion of 1/(1-z) about z=0? about z=a? What is the radius of convergence for each of those series? (hint: use the kind of trick used in class: 1-z=(1-a)-(z-a) = (1-a) [1-(z-a)/(1-a)]. Where and when did we use that trick in class?!)
  2. What is the Taylor series expansion of 1/(1-z)² about z=0 and its radius of convergence? Can you generalize to 1/(1-z)^m where m is a positive integer?
  3. Laurent vs Taylor: Expand 1/z into a Laurent series about z=0 (hint: this is really easy...). Can you expand 1/z in a Taylor series about z=0? What about a Taylor series about z=1? Here's another nice Laurent vs. Taylor: Expand 1/(1+z²) in a Taylor series about z=0 (we've done it in class). Show that it converges only for |z|<1. Now expand in a Laurent series about z=0 (Hint: let z=1/t, t complex and expand the t function in a Taylor series about t=0, i.e. z=infinity(!), the Taylor series in t is your Laurent series in z. Show that the Laurent series converges only for |z|>1. hence the Taylor and Laurent series are complementary. Cool.)
  4. Find the first 3 terms in the Laurent series expansion of 1/sin(z) about z=0. Calculate the integral of 1/sin(z) about a circle of radius 1 centered at the origin.
  5. Calculate the integral of exp(1/z) about an ellipse of major axis 7 in the x direction and minor axis 2 in the y direction, centered at the origin.
  6. Calculate the integral of exp(1/z) and exp(1/z^3) about the unit circle centered at the origin.
  7. Calculate the integral of e^z/[(z²+1)(z+1)²] about a countour that encloses z=-1 and z=i.
  8. Same but for a countour that encloses z=-1, z=i and z=-i.
  9. Calculate the integral of g(z)= z² sin (1/z) over the unit circle. (How about z⁴ sin(1/z) and sin(z²) sin(1/z)?)
  10. Calculate the integral of z⁵ exp(1/z) about the unit circle.
      Hints and explicit solved examples for the following types of integrals are given in section 7.2. which is pretty well written and readable. The first two pages of 7.2 is a summary of the ideas discussed in class Monday Dec 8.
  11. Calculate the integral from -infinity to + infinity of 1/(ax²+ bx + c) where a,b,c and x are real and a>0, b²-4ac < 0. [Hint: same approach as for 1/(1+x²) done in class].
  12. Calculate the integral from -infinity to + infinity of 1/(1+x⁴).
  13. Calculate the integral from -infinity to infinity of cos(x)/(1+x²) and of cos(x)/(1+x⁴) [see exercise 7.2.13]
  14. Given the Poisson integral [integral from -infinity to + infinity of exp(-x²) = sqrt(Pi)], calculate the Fresnel integrals: integral from 0 to infinity of cos(x²) and of sin(x²). Justify! [See problem 7.2.19].
• 7.1 READ! it has important info. Try exercise 7.1.1, we did it in class but do you understand it?
• 7.2 has lots of good, key examples solved explicitly and quite a few good exercises (for a change!), in addition to those "extras" given above for section 6.5 try: 7.2.2, 6-14, 17-19, 22, 23.