Syllabus
The following is a tentative syllabus for Math 60. The actual topics covered will depend upon interest and pace.The rough outline of the course is as follows. Multivariable functions and their derivatives, vector fields, gradient, divergence, curl, double and triple integrals, parametrized curves, flows, line integrals, Green's theorem, the flux integral, Lagrange multipliers, Stokes' Theorem, and the Divergence Theorem.)
Lecture topics
- Week 1
- multivariable functions, functions as mappings, examples, how to visualize: graphs/sections/level sets, examples of surfaces: paraboloids, hyperboloids, etc.
Colley 2.1 - Recap: meaning of limits of m-v functions, examples where limits do
not exist (smooth, singular), what continuity means, examples
including partial derivatives
Colley 2.2,
"differentiable" means good approximation exists, the derivative matrix. example: tangent plane. derivative matrices and meaning of differentiability, example, meaning: best linear approximation in a small nbhd,
Colley 2.3
2nd partials and meaning, mixed partials, when and why equal.
Colley 2.4. - chain rule: tree diagrams, matrix multiplication, directional
derivatives: the gradient, properties of the gradient, properties
of directional derivatives
Colley 2.5, 2.6 - param paths, curves, velocity, accel, circles, ellipses, cycloids, arclength
Colley 3.1 [minus Kepler]
arclength
Colley 3.2
Week 2 -
vector fields, flow lines, grad, div
Colley 3.3 - curl, del operator: examples, defn/meaning
Colley 3.4 - Taylor's Theorem
Colley 4.1, Optimization, critical points, 2nd derivative test
Colley 4.2, Applications of optimization, Lagrange Multipliers.
Colley 4.3. - review cartesian double, triple integrals, meaning, examples, choosing an order of integration, Riemann sum with physical example
Colley 5.1--5.4 - polar, cylindrical, spherical integration, volume elements for each
Colley 1.7, 5.5 [only the special cases of cylinders/spheres!], more cylindrical/spherical examples, which to use?
Colley 5.5 [only the special cases!] - line integrals: scalar/vector, mass of wire, area of fence,
work
Colley 6.1 - Green's Theorem
Colley 6.2,
conservative vector fields, fundamental theorem of line integrals
Colley 6.3 - surface integrals
Colley 7.2 - Stokes Theorem
Colley 7.3 - Divergence Theorem, unifying theorems
Week 3