Question

Find an equation of the tangent plane to the given parametric surface at the specified point.

r(u, v) = u cos vi + u sin vj + vk; u = 9, v = p/3

Answer 1

Let S be the helicoid (or spiral
ramp) with the vector equation
r(u; v) = u cos vi + u sin vj + vk; 0 u 1; 0 v :
Then
ru(u; v) = cos vi + sin vj;
rv(u; v) = ??u sin vi + u cos vj + k;
and
ru(u; v) rv(u; v) = sin vi ?? cos vj + uk:
Hence the area of S is given by
A(S) =
ZZ
0u1; 0v jru(u; v) rv(u; v)j dA
=
ZZ
0u1; 0v
q
sin2 v + (??cos v)2 + u2 dA
=
ZZ
0u1; 0v
p1 + u2 dA
=
Z 1
0
Z
0
p1 + u2 dv du
=
Z 1
0
p1 + u2 du
=

u
2
p1 + u2 +
1
2
ln

u + p1 + u2
1
0
=

2
hp2 + ln

1 + p2
i
: