Last time: Project Day!
Today:
- Great job with projects! We enjoyed them, and hope that you enjoyed each
others' work, too.
- Return 17.6 problems, and 17.7a problems.
- Collect problems 17.7b, 17.8.
- Problems for 17.9 will be accepted with final. Both 17.8 and 17.9 are
"optional", and will only serve as replacements for missing homeworks, or
homeworks with lower grades....
- Questions/Comments on old stuff?
- The final:
- 5/5, 1:00-3:00. Contact me with questions up
to the final:
- You'll be allowed a 3x5 card again.
- The exam will be 50% old stuff, 50% new
- For the new stuff (chapter 17, and section 16.9), you'll
not be allowed to skip any problems. It will be a
sample, of these sections: I can't get complete
coverage, due to the time constraints.
- For the old stuff, you'll be allowed to skip a couple of
problems.
- Remember: I'll be available to work with you tomorrow night,
Tuesday, from 9:00pm until you give up or we all fall asleep.
- Material:
- General observations:
- How often did a concept in multivariate calculus
turn out to be some simple generalization of a
univariate concept? Often! It's just like it was Back in Calc I.
- Linear things continue to be extremely important
(tangent plane approximations) -- because
they're easy to work with.
- Variants of integration are always doing the same
job: adding up a bunch of tiny little things
(infinitesimals) to get some finite, ordinary
answer....
- 15.1: Multivariate Functions: domains, ranges, defined in
different ways, linear functions, level surfaces,
contour plots, "legal" coordinate systems.
- 15.2: limits, continuity, multiple methods of approach;
discontinuities and level curves; polynomials and
rational functions.
- 15.3: partial derivatives (holding other variables fixed,
as if they were constant); partial derivatives as
limits; higher derivatives; Clairaut's theorem
(equivalence of mixed partials).
- 15.4: Tangent plane approximations to a surface;
linearization of function f; differentiability
implied by existence and continuity of partials.
- 15.5: the chain rule; tree diagrams and directed graphs
help keep track of the bookkeeping.
- 15.6: the gradient (direction of fastest increase,
perpendicular to level curves); directional
derivatives.
- 15.7: critical points; extrema (maxima and minima); using
the discriminant D to distinguish the three
cases of extrema; strategy for finding extrema.
- 15.8: Lagrange Multipliers; useful for finding extrema on
a boundary; optimizing subject to constraints.
- 16.1: volumes rather than areas; Riemann sums and
approximation methods; integration is still
linear.
- 16.2: Iterated integrals over rectangular regions;
Fubini's theorem; separable integrals.
- 16.3: Double integrals over general regions (variable
limits of integration).
- 16.4: Double integrals in polar coordinates. "The pirate's
dA" (r-dr-d-theta).
- 16.5: applications of integrations: moments of inertia,
center of mass, radius of gyration, joint density
functions, expected values.
- 16.6: Surface area; the cross-product of the tangent
parallelogram patch yields differential area.
- 16.7: Triple integrals (no big news); Fubini's theorem
still works, no surprise; different "types" of regions
lead to different orders of integration.
Applications. Drawing the danged things!
- 16.8: Triple integrals in cylindrical and spherical
coordinates; various differential volumes dV.
- 16.9: Change of variables in multiple integrals;
one-to-one transformations; the Jacobian as an
expansion/contraction.
- 17.1: Vector fields (e.g. gradient fields for scalar
functions); velocity fields; conservative vector
fields and potential functions; visualizing vector field.
- 17.2: line integrals along curves; parameterizations; line
integrals with respect to arc length, x, or
y; orientation; line integrals of vector fields
(e.g. work calculations from physics).
- 17.3: Fundamental theorem for line integrals; independence
of path; tests for conservative vector fields; spotting
a non-conservative vector field from a graph.
- 17.4: Green's theorem; positive orientation; a simple (?)
generalization of the fundamental theorem of calculus.
- 17.5: Curl and divergence; curl is rotation, divergence
compressibility; curl(grad f)=0; conservative
vector fields have zero curls; div(curl F)=0;
variants of Green's theorem (Stokesish, and
Divergenceish - using normal component).
- 17.6: Parametric surfaces (require two parameters);
patches in parameter space mapped to surfaces embedded
in three-space; special parameterization for surfaces
of rotation; tangent planes of parameterized surfaces;
surface area.
- 17.7: Surface integrals; those cross-products (weighted)
again!; surface orientation; flux; surface integrals of
parameterized surfaces.
- 17.8: Stokes's theorem; line integrals to surface
integrals (and back!); circulation.
- 17.9: Divergence theorem; surface integrals to volume
integrals (and back!); sinks and sources.
- Chapter 17 summary
- A semester's
worth of gnuplot plots
Next time: Final Exam!
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