Part III (Double and triple integrals)

15.1: Not much here, just make sure you think of y as a number while
      you do a dx integral (and conversely).


      Of course, know Fubini.


15.2: I am going to write
      	\int_a^b \int_u(x)^v(x) F(x,y)dy dx
	and then "\int" is an integral sign, a and u(x) are lower
      	bounds, and b and v(x) upper bounds.

      The thing to remember is that R is in the xy-plane, and you
      (typically) have to decide whether you make slices parallel to
      the x- or the y-axis. If you slice R parallel to the y-axis,
      then in the inner integral you have y running, and in the outer
      one x. So your integral looks like dy dx.
      This requires you to make sure that the upper and lower
      limits in the inner integral are functions of x, or plain
      numbers, (you don't want to have any y around anymore after the
      inner integral) and the bounds for the outer integral are just
      numbers (like in the one I wrote above). 

      For a dx dy integral it is the other way round, the inner bounds
      should be functions of y but not x.

      If it comes to computing a volume that lives over the 2D region
      R inside the plane, think of your slices cutting R into little
      squares. Then in 3D, over each square stands a column of
      material, and you want its volume inasmuch as it is inside the
      solid whose volume you want. Pretending this column meets the
      inside of your solid in a rectangular prism of dimensions (dx)
      times (dy) times (height), the height comes about as the
      difference between the highest point of that column that is
      inside the solid, minus the lowest point that is still in the
      solid.

      You should know the basic rules how to calculate with double
      integrals on page 907.

      Sample problems: 9,12,20, 28, 39,44,52,58,60,63,71,75,78

15.3  Sample problems: 18, 20,23

15.4  Know how to go back and forth between polar and Cartesian
 
      Sample problems: 5, 6, 12, 19, 23, 26, 29, 32, 39, 43

15.5  The main difficulty is setting up the integrals (finding order
      of integration that is best, and then setting the limits).
      This can only be improved with practice. Remember that you may
      want to use polar cordinates sometimes for solving (or even for
      setting up) the integrals. Know how to rewrite a dx dy dz
      integral as a dy dz dx integral, as well as the other
      variations. (The relevant skill is to determine the new integral
      limits.) 

      Sample problems: 3, 5, 14, 21(all parts), 25, 35, 38, 41, 46

15.6  Know the definitions/formulas of mass, 1st moment (leverage),
      center of mass/centroid, and 2nd moment (moment of
      inertia). Know the Theorem of Parallel Axes.

      Know what is a joint probability density function, and how to
      recognize one. Know how to compute means/expected values.

      Sample problems: 6, 11, 14, 22, 25, 36, 40, 43

15.7  Know the conversion formulas Cartesian <--> cylindrical.
      Know the conversion formulas Cartesian <--> spherical.
      Have an idea when to use Cartesian/spherical/cyclindrical.

      Sample problems: 5, 10, 17, 18, 25, 26, 29, 35, 37, 45, 51, 56,
      64, 65, 70, 73, 80, 86, 91, 102.

15.8  Know the definition of the Jacobian, and have some practice in
      changing coordinates in double/triple integrals.

      Sample problems: 2, 6, 9, 14, 20, 23, 40 on page 965


16.1  Know the definition of a line integral for a scalar function
      f(r).

      Know the basic properties (additive in f, homogeneous,
      cumulative in the path) and that it can change if you change the
      path. Know the area interpretation if the path is in the xy plane.

      Be able to do mass and moment calculations on coils, thin wires,
      etc. with density function delta.

      Sample problems: 1 through 8, 14-16, 34, 39,

16.2  Know what is flux, flow, work, circulation, line integrals of vector
      fields (tangential and normal) and how to evaluate them.

      Know the conventions for writing Mdx+Ndy+Pdz and integrals of this.

      Know+understand the table on page 982 on work= \int F*dr= \intF*T ds.

      Understand \int F*T ds as a flow if F is not a force field but a
      velocity field.

      Flux is measured against the normal via \int F*n ds.

      Know the explicit formula for flux, page 985. (Or know how to get
      it from basic principles, which is better).

      Sample problems: 12, 17, 20, 29, 35, 36. 47, 48, 49, 59, 62

16.3  Know: conservative field, potential function, simply connected.

      You should be able to find potentials of conservative fields,
      and know how to check a field for conservativity. (Component
      test).

      Know: exact/closed differential form, and the component test for this.

      Sample problems: 1,2,3, 7,8,9, 14, 20, 25, 30, 33, 34, 38.

16.4  Know circulation density (curl) and flux density (divergence) of
      a plane vector field.

      Know and understand both versions of Green's theorem, the one
      with
		curl on R <-->circulation along C
      and the one with
       	        divergence on R <-->flux across C.

      Know how to use Green to get areas.

      Be able to recognize simply connected/not simply connected
      regions in the plane.

      Sample problems: 1, 2, 12, 23, 26, 28, 32, 38, 44, 45, 48.

16.5  Have some practice in parameterizing surfaces. Know meaning of
      "smooth". 

      Know how to find area of surfaces, especially of a graph surface.

      What is an implicit surface? How do you find its area? Know the formula.

      How do you find tangent planes in all these cases?

      Sample problems: 2, 6, 12, 14, 18, 24, 30, 31, 32, 40, 50, 56.

16.6  Know the three forms for surface integrals on page 1023 (for
      parametric, implicit, explicit surfaces).

      What is an orientation?

      Know surface integrals of vector fields for: parametrized surface,
      level surface.

      Use your understanding of what first and second moments mean to
      be able to reproduce their definitions for surfaces. (Of course,
      you can learn them by heart as well, but that takes memory that
      could be used otherwise).

      Sample problems: 2, 4, 13, 14, 15, 25, 28, 33, 34, 37, 44, 48.

16.7  Know curl in 3D. Know Stokes (with some understanding of the
      hypotheses needed). Know identity (8) in that section. Know the
      implications mapped out on page 1043.

      Sample problems: 1-6, 7, 13, 17, 22, 27, 31, 34.

16.8  Know div F. Know the flux-divergence Gauss theorem, with some
      understanding of the hypotheses needed.

      Know the identity in Thm 9.

      Know the 4 identities on page 1055 and its meta-paraphrasing at
      the bottom of that page.

      Sample problems: 1-4, 11, 12, 22, 26, 28

      Read through problems 35, 36 to gain appreciation for how
      mathematical theorems infom you on physical differential
      equations. 

      Have some idea how to use Gauss/Green/Stokes for simplifying life.

From the practice exercises chapter 16: 3, 11, 16, 21, 27, 29, 30, 37,
      39, 41, 44, 51, 53, 59.

From the Additional and Advanced Exercises Chapter 16: 3, 9, 13, 16.