Generalities. Our final exam will be given on Thursday,
December 18, from 9:00 a.m. to 12:00 a.m.. It will cover everything
we learned in class this semester, up to and including Stokes'
Theorem.
Preparation. While this exam will be in the same spirit as the
midterms, the homework, and the lecture (in terms of the material
required), it will not consist of a rehashing of those problems and
examples. However, if you have been allocating a respectable amount
of time for working on the homework, attending class regularly and
staying up to speed on the material, you should be ready for the exam.
I'll be holding a review session (day/time TBA, probably Wednesday
evening). Additionally, you can make use of the posted practice
problems if you're feeling rusty on a particular section, or if you
would otherwise want to practice.
Bare Essentials. There are certain formulae, techniques and
definitions that I will assume you know well by now. You should be
able to summon them on command, and so you might be short on time if
you spend exam minutes trying to remember those basics (instead of
using them on the problem at hand). They include (this is mostly cut
& pasted from the midterm review stuff):
3-D basics. Plotting points, finding distance between
points, finding equations of spheres, and being able to recognize
equations of spheres (including completing the square, etc, to find
the center and radius).
Vector basics. Vector notation using angle-brackets or
i, j and k. Algebraic and geometric aspects of:
adding two vectors, multiplying a vector by a number, norm, the dot
product (and the dot product formula), the cross product (and the
cross product formula), parallel and perpendicular vectors, finding
vectors given endpoints. Illegal operations (e.g. vector + number).
Planes. Finding the equation of a plane given a point and a
normal vector, a point and two vectors parallel to the plane, three
non-collinear points contained in it, two intersecting lines contained
in it, a point and a parallel plane, a point and a perpendicular line,
etc.. Recognizing the equation of a plane (especially the normal vector).
Lines. Finding the (parametric, symmetric) equation of a
line given a point and a parallel vector (or line), a point and two
perpendicular lines, a point and a perpendicular plane, two
intersecting planes, etc.. Finding the intersection of two lines (if
they intersect), and more generally, intersections of lines with
planes, spheres, and other surfaces.
Vector Functions. The definition of a vector function and
how it generalizes the parametric equation of a straight line.
Derivatives of vector functions algebraically and geometrically.
Formulae/definitions for the velocity, speed, acceleration, unit
tangent vector, normal vector, curvature. Arclength.
Multivariable Functions. Generalities (convince yourself of
what a domain and a graph look like ...), level curves and how they
help in understanding what a graph looks like. Partial derivative
computation and geometric interpretation.
Partial differentiation techniques. Computing partial
derivatives of functions of several variables. The chain rule. What
the partials mean. Using the partials to determine the min's and
max's of a given function, and to compute the directional derivative
along a given direction (or to find a specified direction etc).
Tangent planes. Finding the equation of a tangent
plane to a surface at a given point, using both the three-dimensional
gradient approach, and, in case the surface is the graph of a function
of two variables, the "standard" formula. Understanding why these two
formulas are the same.
Gradients. More generally, understanding the properties
of the gradient (for instance, the gradient of a function of two
variables is a two dimensional vector that lives in a horizontal
plane, is perpendicular to the level curve, and points in the
direction of maximal ascent).
Lagrange multipliers Using the method of Lagrange
multipliers to find the min's and max's of a given function with a
given constraint.
Integration. Computing the double integral of a function of
two variables over a given region (the region could be given in
different ways). Recognizing the region if the endpoints are given.
Switching the order of integration and computing the new endpoints.
Polar Integration. Spotting when an integral is better
suited for a polar change of coordinates. Setting up polar integrals
(including the endpoints, and the appropriate variable substitution).
This might also require knowledge of polar graphing techniques.
Line Integrals. Computing line integrals of functions with
respect to arclength, and with respect to x or y.
Computing arclength. Computing line integrals of vector fields.
Determining when a vector field is conservative/path independent, and
using the Fundamental Theorem of Line Integrals. This might require
knowledge of parametrizing curves.
Green's Theorem. Using both "sides" of Green's Theorem in
integration computation. Using Green's Theorem to find areas.
Understanding the relation between Green's Theorem and the Fundamental
Theorem of Line Integrals.
Surface Integrals. Computing surface integrals of
functions with respect to surface area. Computing surface area.
Computing surface integrals of vector fields. This might require
knowledge of parametrizing surfaces.
Stokes' Theorem. Using both "sides" of Stokes' Theorem in
integration computation. This requires knowing how to compute the
flux of a vector field.
What else to expect. As you've seen already, my exams tend to
be a bit on the long side, so you should try to make it to class on
time that day. Second, make sure you understand the homework problems
(even if you got them right!). Third, there are no make-up exams, and
I do not accept excuses after the fact. That is, if you are having
trouble with the material for whatever reason, this is the time to try
tackling it (though I prefer not to rehash the entire class on the
morning of the test). And finally, if you tend to stare into space
while thinking about an exam problem, make sure that you are not
looking in the direction of a fellow student's paper.
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