3.3 Curves on Surfaces
3.3.1 Parametric curves in the
–
–plane
Definition: A curve in the
–
–plane is a map
which associates with each values of the
parameter
(‘time’
) a point
in the
–
–plane.
Examples
1. The circle around the origin,
Check that
for all
.
2. The curve line
Sketch this!
3.3.2 Parametric urves on Surfaces
Consider a function
, i.e. a surface
above the
–
–plane. Consider a curve
in the
–
–plane. This curve defines a corresponding
curve
on the surface. Example: for
and
,
. The circle in the
–
–plane corresponds to a ring hovering at a
distance
above the plane, being part of the surface of the
paraboloid
. Sketch the corresponding picture (lecture)!
3.3.3 Change of height along a Curve
Reminder: Total change of height (total
differential)
is called the
total differential of
at the point
and gives the total change of the height of the surface
(measured from the
–
–plane) at the point
, if we move a tiny step
along the
–direction and a tiny step
along the
–direction.
From this, we can calculate the change of the
height of the curve
:
This is a
chain rule
Example:
and
We have
We can check this by direct calculation,
. The general formula, however, makes it clear that
there a two contributions to the change of the curve
on the surface: 1. the ‘geometric change’
(partial derivatives
,
) of the surface. 2. the ‘kinematic change’,
i.e. the time derivatives
,
that determine the speed by which we sweep along the
curve
.
Example:
and
Direct check with
.