and
are
well-defined everywhere, but
is well-defined everywhere except at the origin (0, 0, 0).
x = t2
y = t 3
z = 0
Therefore the equations of the curve are
“bounces” from
to
abruptly
at t = 0
(and also changes
abruptly at t = 0, from
to
)
Everywhere except t = 0,
For the surface whose equation in cartesian coordinates is
z2 = x2
- y2
Þ
-
x2 + y2 +
z2 = 0
This quadric surface is of the type
with a = b
It is therefore a
circular cone |
---|
whose axis of symmetry is the x-axis
At (1,1,0)
(Any non-zero multiple of the gradient vector is a normal
vector.)
A point on the tangent plane is (1, 1, 0)
Þ the tangent
plane is
or
Normal line:
or y = 2 - x,
z = 0
[Note that the line y = x, z = 0
is one of the generators of the circular cone.]
For the surfaces whose cartesian equations are
z = 3x2 + 2y2
and
z = 3x2 + 2y2
is of type
elliptic paraboloid
with its axis of symmetry along the z-axis.
-2x + 7y2
- z = 0
Þ
7y2 = +2x + z
is of type
parabolic cylinder
Its vertex line is 2x + z = 0,
y = 0
[One should check that (1, 1, 5) does indeed lie on
both surfaces]
q = Angle
between surfaces = Angle between
Calculate the directional derivative of
at the point
Given that the velocity of a particle at any time
t is
and that the particle is at the origin when
Therefore
everywhere along the curve
= constant
Þ
the curve lies in one plane
The plane normal
= any non-zero multiple of
.
Choose
The curve passes through the origin
The equation of the plane is
[Note: the curve is a parabola in this plane.]
Alternative Calculation of Unit Principal Normal :
Use the acceleration vector:
But