Department of Mathematics
University of Southern California
February 25, 2011
Tangent Plane/Gradient Notes
For a function
F
=
F
(
x, y, z
), when it is held constant, say,
F
(
x, y, z
)=
k,
it gives a relationship between
x
,
y
,and
z
. This may be viewed as
z
(
x, y
), or
x
(
y,z
),
or
y
(
z,x
), and forms a surface. Such a family of surfaces for various values of
k
is
called level surfaces. Now consider an arbitrary curve con
f
ned to a particular level
surface. Such a curve may be described parametrically as
r
(
t
)=
<x
(
t
)
,y
(
t
)
,z
(
t
)
>
Based on our earlier work on three-dimensional curves, we know that
r
I
(
t
)=
<x
I
(
t
)
,y
I
(
t
)
,z
I
(
t
)
>
is a tangent to the curve. Now, we are considering the surface
F
(
x, y, z
)=
k
on which
r
(
t
) lies. If we remain con
f
ned to the curve (which is con
f
ned to the level surface),
we may express the curve as
F
(
x
(
t
)
,y
(
t
)
,z
(
t
)) =
k.
Now, di