Arc Length:
Be able to compute the arc length of a vector valued function, as the
integral
.
Know what it means for a curve to be
parameterized by arc length, and given a parametric curve,
reparameterize it by arc length.
Tangent, Normal, and Binormal vectors:
Given a parametric curve R(t), be able to calculate the unit tangent
vector
,
the unit normal
,
and the unit binormal
.
Given those, you should be able to find the normal plane to
the curve at a given point, and the osculating plane (the latter is
the plane with B(t) as a normal vector.
Motion in space:
If the position of a particle is given by R(t), its velocity vector
is
V(t) = R'(t), and the acceleration is given by
A(t)=V'(t)=R''(t).
Parametric Surfaces:
Be able to match a surface to its parametric representation, and to
describe how a parametric surface changes as its domain is adjusted.
Level Curves and Contour plots:
Understand what level curves are, what a countour plot is, and their
relationship to the graph of a surface.
Limits and Continuity:
Be able to compute limits of a function of two variables, and be able
to determine when a function of more than one variable is continuous.
Partial derivatives:
Be able to compute partial derivatives and second partials. Understand
what they represent. Know how to use the chain rule to compute
partial and total derivatives of compositions.
Tangent planes, linear approximations:
Be able to find the tangent plane to a surface, and use it to
approximate a function.
The gradient vector; directional derivatives:
Know what the gradient vector is, and its properties. For example,
that
points in the direction of greatest increase of f,
and that
is always perpendicular to level curves of f.
Understand what the derivative in the direction of a unit vector u
is, and how to compute
.
Critical points, Maxima and Minima:
Be able to locate critical points of functions of several variables.
Be able to use them to locate local maxima and minima. Also be able
to find extreme values (i.e. absolute maximum and minimum values) of
functions defined over closed, bounded regions of the plain. Use the
second derivative test where necessary.
Definition of double integrals:
Understand the definition of the double integral. Specifically, be
able to do problems like those assigned in section 12.1.