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1. Given the following five vectors:
.
Do each of the following:
a) Form the sum:
.
b) Compute
.
c) Compute
.
d) Find values for
and
for which
and
.
e) Find the cosine of the angle between
and
. Between
and
(the answer will be a function of t).
f) Find the projection of
on
.
g) Find the determinant whose columns are
and
; also find the determinant whose columns are
and
.
h) Suppose the point
has coordinates
. What are its spherical coordinates
and
?
i) What is the volume of the parallelepiped with edges
and
?
j) Find the projection of
into the
plane. What is its length?
2. Consider the line containing the points
and
above
.
a) Give a parametric representation of the points on that line.
b) Find a unit length "tangent vector" that points in the direction of the line.
c) Find two directions normal to that vector.
d) e) and f) consider the plane containing the points
and
:
Find a (two parameter) parametric representation of the plane.
Find a normal to the plane.
Find an equation that points on the plane all obey.
g) Suppose we have a new and different product of vectors
that has the property
for all
and
is linear in each argument so that you can apply the distributive law.
Deduce something about
by applying same to
.
3. Differentiate the following functions with respect to the indicated variables:
a)
.
b)
with respect to
for fixed
.
c)
with respect to
for fixed
.
d)
with respect to
everything else fixed.
e) Find the gradient of
.
f) Find the directional derivative of this function in the direction whose unit vector is
.
g) Find the linear approximation to
at
.
h) Evaluate the derivative with respect to
of
where
is
; suppose that
is in the direction of
. What then is the answer?
i) Where is
not differentiable? Where is
not differentiable? Where is
not differentiable?
j) Find the derivative of an inverse function to
(to define an inverse function completely you have to specify a range; ignore that here).
4.
a) Find the gradient of the function
and
.
b) Find the gradient of
.
c) Find the gradients of
and of
.
d) Find the curl of
.
e) Find the divergence of
(remember that
).
f) Find the curl of same.
5.
a) Find the quadratic approximation to
at
(radians).
b) Where does this function have critical points (both partial derivatives are 0).
c) Find at least one saddle point.
d) Evaluate
by switching a dot and cross product and expressing the triple cross product according the rule for doing same, to get an alternate expression for the same thing entirely in terms of dot products.
e) Which of the following functions can be defined at
?
?
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