1. Consider the surface
S =
_x005f (x, y, z) ∈ R
3
: x
2 + y
2 + z
2 = 2 .
Let C1 denote the curve given by the intersection of S with the surface
S1 =
( (x, y, z) ∈ R
3
: z =
r
x
2 + y
2
2
)
.
Let also C2 be the curve given by the intersection of S with the surface
S2 =
(x, y, z) ∈ R
3
: x
2 + (y 1)2 + z
2 = 1, z > 0
.
Please complete the following calculations and derivations without using any
decimal representations (i.e., keep any numbers arising as integers, fractions,
radicals or function evaluations thereof).
(a) Give a brief description of S1 and S2 (e.g., circular cylinder, radius equal to 1).
(b) Write down Cartesian representations for C1, C2. In each case, identify the curve as
a certain standard planar curve (provide a brief description).
(c) Find the point P of intersection of C1, C2 with positive coordinates.
(d) Using the descriptions from part (a), write down parametrisations for C1, C2. In each
case, find the parameter value corresponding to the point P from part (b).
(e) Find the tangent t1 to C1 at P. Find also the tangent t2 to C2 at P.
(f) Find a vector u perpendicular to both t1 and t2 at P and check that it is parallel to
the normal direction to the surface S at P.
(g) Does u point away from the interior of S Justify your answer
