Math 114 log
Section 001 Spring 2015
August 26, 2015: 12.1-12.3, and definition of cross product.
August 28, 2015: 12.4 and 12.5.
Stuff not in the textbook: cross product is
invariant under all rotations in dimension 3, approach toward showing
the formula for the cross product (u x v) x w of
three vectors.
August 30, 2015: 12.5 (lines and planes in 3-space), beginning of 12.6
(cyliners, aka ruled surfaces), hyperboloid of one sheet.
Stuff not in the textbook: the family of all lines in the plane form
a 2-dimensional formula. (In the language of the 19th century,
one would say that "the number of moduli for lines in the plane
is 2".
Septermber 2, 2015: 12.6, how to tell two-sheeted hyperboloids from
one-sheeted ones, How describe a solid in 3-space by its projections
to planes and lines; 13.1, vector-valued functions as curves in 3-space.
September 4, 2015: 13.1 and 13.2, tangent vectors to curves,
arclength (which is invariant under reparametrization).
September 9, 2015: 13.3, product rule, parametrization by arc length,
unit tangent field to a curve, definition of the principal normal
of a curve and the curvature.
September 11, 2015: 13.4 and 13.5, definitions of curvature, torsion,
tangent, principal normal and binormal vector fields, the
Frenet-Serret formulas.
September 14, 2015: 13.4 and 13.5: review the Frenet-Serret frames
for curves in space, example of computing the tangent, normal,
binormal, curvature and torsion.
September 16, 2015: 13.5 (wrap up), 13.6, plane curves in
polar coordinates, Kepler's 2nd law.
September 18, 2015: 14.2 and 14.3, definition and example of limit
of functions in several variables, linear approximation and
partial derivatives.
September 21, 2015: 14.3-14.5. Recalled the idea of linear
approximation as applied to the definition of differentiability,
partial derivatives, directional dirivatives. The chain
rules comes from composition of linear approximations.
September 23, 2015: Defined the Jacobian matrix for a matrix-valued
function, explained the chain rule in terms of the Jacobian
matrices (section 14.4).
(Did not employ/explain the mnemonic "branch diagram" as employed
in the book.)
September 28, 2015: 14.5, beginning of 14.7, 14.9,
directional derivative, mixed partial derivatives,
Taylor expansion of functions in several variables, critical point.
September 30, 2015: 14.7, use Taylor expansion up to second order
terms to determine whether a critical point is a local max/min;
how to do this in the case of two variables.
October 2, 2015: 14.8, Lagrange multiplier
October 7, 2015: Lagrange multipliers (general case), critical
points under constraints, singular points.
(The upshot is: local extrema of a function f occur at critical points
(under constraints if contraints are present), or
boundary or singular points for the constraints.
October 12, 2015: 15.1,15.2, 15.3, definition of multiple integrals,
Fubini theorem.
October 14, 2015: examples of multiple integrals, change of variable
formula
October 16, 2015: 15.5, 15.7
change of variable formula illustrated, evaluation of the
improper integral of the Gaussian distribution,
polar coordinates, cylindrical coordinates.
October 19, 2015:
15.7, spherical coordinates. Use spherical coordinates to compute
the volume of a spherical solid.
October 21, 2015:
part of 16.1 (definition of unsigned line integras),
16.5 and part of 16.6 (definition of unsigned surface integrals).
October 23, 2015:
Unsigned surface integrals, review definition
and work out some examples, including (a) area of a sphere,
(b) center of mass of a half-sphere and (c) area of a torus.
October 26, 2015:
Overview of the fundamental theorem of calculus,
orientation of curves and surfaces.
October 28, 2015: Orientation of surfaces continued,
the sign of a parametrization,
how to compatibly orient a surface with boundary and its boundary,
November 4, 2015:
definition of oriented/signed surface integrls + an example,
definition of oriented/signed line integrals.
November 6, 2015:
The statement of the fundamental theorem of calculs:
Stokes' theorem and the divergence theorem,
the definitions of curl and divergence.
November 16, 2015:
Worked out, in detail, the line integrals of the vector field
F:= -y/(x^2+y^2) i + y/(x^2+y^2) j
over serveral closed curves on the plane, using serveral
methods, including the Stokes/divergence theorem.