1
Math E-21a Mega-List of Things You May Want to Know – Fall 2009
1. Write equations for surfaces in
R
3
that are characterized geometrically by distances (spheres, cylinders,
etc.), and determine their intersection with specified planes.
2. Sketch or identify contour plots of real-valued functions on
R
2
. Identify plots of function graphs for such
functions.
3. Given vectors in
R
2
or
R
3
, do addition, scalar multiplication, dot and cross products.
4. Manipulate vector expressions symbolically (distributive law, triple product,
A
(
B
C
)).
5. Express lengths, angles, areas of triangles and parallelograms, and volumes of parallelepipeds and
tetrahedra in terms of vectors. Identify and construct orthogonal vectors.
6. Find the scalar and vector projections of a vector in any given direction.
7. Calculate the intersection (if any) of specified lines and planes. Resolve a vector into components parallel
and perpendicular to a specified vector, line, or plane.
8. Calculate the distance from a point to a specified line or plane, or between two nonintersecting lines in
R
3
.
9. Calculate partial derivatives and directional derivatives of functions of two or more variables.
10. Calculate the gradient of a real-valued function of two or three variables, and state and apply the relation
between gradients and level curves or surfaces.
11. Calculate the rate of change of a real-valued function along a parametrized path.
[Basic Chain Rule:
(()
, ()
)
fd
x
y
d
f xt yt
f
dt
xd
t
yd
t
v
for a path in
R
2
;
,()
)
x
y
z
d
f xt yt zt
f
dt
x dt
y dt
z dt
v
for a path in
R
3
.]
12. Given a surface in
R
3
described by the graph of a function
f
(
x
,
y
), find the equation of the tangent plane at
a point on the surface where
f
is known to be differentiable, and use it to find approximate values for this
function near the point of tangency.
[Linear approximation:
00
0
0
(
,
) (,) (,)
(
) (,)
(
)
xy
f xy
f x y
f x y x x
f x y y y
]
13. Given a function
s
=
f
(
u
,
v
) or
s
=
f
(
u
,
v
,
w
) whose arguments
u
,
v
,
w
are specified functions of
x
,
y
, and
perhaps
z
, use the chain rule to derive or verify relationships among the partial derivatives of
s
with
respect to
x
,
y
, and
z
.
14. Given functions that express Cartesian coordinates
x
and
y
in terms of other coordinates
u
and
v
, and a
function
z
=
f
(
x
,
y
), use the chain rule to express partial derivatives of
z
with respect to
u
and
v
in terms of
partial derivatives of
f
with respect to
x
and
y
.
15. Given a curve in the plane specified by
f
(
x
,
y
) = constant, use implicit differentiation to find a formula for
the derivative of the function that specifies
y
in terms of
x
near a specified point on the curve, and use the
value of this derivative to determine a tangent line or to do a linear approximation near the point.