Day 45 in Mat329

Your exam will be half old material, half new.

Old material will resemble material from your first two exams. New material will resemble material from your homeworks (or, in the case of 16.5, the problems presented in class).

• How often did a concept in multivariate calculus turn out to be some simple generalization of a univariate concept? We generalized old friends:
  • graphs,
  • limits and continuity,
  • tangent lines (and linearization),
  • the derivative (both the limit definition and the gradient),
  • quadratics,
  • numerical integration techniques,
  • integration (reducing multiple integrals to a succession of univariate integrals, or generalizing the fundamental theorem),
  • The fundamental theorem of calculus, etc.
  (and made some new ones, I hope!).

• Symmetry is perhaps my favorite mathematical idea: it is so powerful, and saves us work. I love to save work! I hope that you do, too.

• 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....

  It's just that the "things" get more interesting in higher dimensions.

• 14.1: Multivariate Functions: domains, ranges, defined in different ways, linear functions (the most important functions -- still -- in calculus), level surfaces, contour plots, "legal" coordinate systems. Drawing (and visualizing) functions. Defining functions (even functions whose values are politicians, or sodas). Some we can do calculus on, and some we can't!
• 14.2: limits, continuity, multiple methods of approach (even non-linear); discontinuities and level curves (old styles, e.g. holes, and new styles, e.g. cliffs -- existing along one line (or curve) of approach, but not another); polynomials and rational functions, trig and exponential -- continuous on their domains. $\epsilon-\delta$ definition of a limit.
• 14.3: partial derivatives (holding other variables fixed, as if they were constant); partial derivatives as limits -- extending "the most important definition in calculus"; higher derivatives; Clairaut's theorem (equivalence of mixed partials, given continuity of the mixed partials -- hence, the importance of continuity).
• 14.4: Tangent plane approximations to a surface; linearization of function f; existence and continuity of partials implies differentiability.
• 14.5: the chain rule; a lot of bookkeeping!
• 14.6: the gradient (direction of fastest increase, perpendicular to level curves); directional derivatives as dot products of vector-directions and the gradient.
• 14.7: critical points; extrema (maxima and minima); using the discriminant D to distinguish the three cases of extrema; strategy for finding extrema, including the Extreme Value Theorem (checking boundary).
• 14.8: Lagrange Multipliers; useful for finding extrema on a boundary; optimizing subject to constraints; generalizing a problem to find a solution; using the gradient.

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• 15.1: volumes rather than areas; Riemann sums and approximation methods (midpoint and trapezoidal masks); integration is still linear.
• 15.2: Iterated integrals over rectangular regions; Fubini's theorem; separable integrals take us right back to univariate techniques.
• 15.3: Double integrals over general regions (variable limits of integration). The hardest part is wrapping your head around the slices.
• 15.4: Double integrals in polar coordinates. "The pirate's dA" (r-dr-d-theta). Things with circular symmetry.
• 15.7: 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!
• 15.8: Triple integrals in cylindrical and spherical coordinates; various differential volumes dV. Things with spherical symmetry.

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• 16.1: Vector fields (e.g. gradient fields for scalar functions were our first introduction); velocity fields; conservative vector fields and potential functions; visualizing vector field.
• 16.2: 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).
• 16.3: Fundamental theorem for line integrals (generalization of the fundamental theorem of calculus); independence of path; tests for conservative vector fields; spotting a non-conservative vector field from a graph.
• 16.4: Green's theorem (interchangeability of two integrals, a line and area integral pair); positive orientation; another generalization of the fundamental theorem of calculus: you can learn about stuff on the interior from the boundary of the interior.
• 16.5: Curl and divergence; curl is rotation, divergence compressibility; $curl(grad f)={\bf{0}}$; conservative vector fields have zero curls; div(curl F)=0; variants of Green's theorem.