4Surfaces and surface integrals
IA Vector Calculus
4.1 Surfaces and Normal
So far, we have learnt how to do calculus with regions of the plane or space.
What we would like to do now is to study surfaces in
R
3
. The first thing to
figure out is how to specify surfaces. One way to specify a surface is to use an
equation. We let
f
be a smooth function on
R
3
, and
c
be a constant. Then
f
(
r
) =
c
defines a smooth surface (e.g.
x
2
+
y
2
+
z
2
= 1 denotes the unit sphere).
Now consider any curve
r
(
u
) on
S
. Then by the chain rule, if we differentiate
f(r) = c with respect to u, we obtain
d
du
[f(r(u))] = ∇f ·
dr
du
= 0.
This means that
∇f
is always perpendicular to
dr
du
. Since
dr
du
is the tangent to
the curve,
∇f
is perpendicular to the tangent. Since this is true for any curve
r(u), ∇f is perpendicular to any tangent of the surface. Therefore
Proposition. ∇f is the normal to the surface f(r) = c.
Example.
(i)
Take the sphere
f
(
r
) =
x
2
+
y
2
+
z
2
=
c
for
c >
0. Then
∇f
= 2(
x, y, z
) =
2r, which is clearly normal to the sphere.
(ii)
Take
f
(
r
) =
x
2
+
y
2
− z
2
=
c
, which is a hyperboloid. Then
∇f
=
2(x, y, −z).
In the special case where
c
= 0, we have a double cone, with a singular apex
0. Here ∇f = 0, and we cannot find a meaningful direction of normal.
Definition
(Boundary)
.
A surface
S
can be defined to have a boundary
∂S
consisting of a piecewise smooth curve. If we define
S
as in the above examples
but with the additional restriction
z ≥
0, then
∂S
is the circle
x
2
+
y
2
=
c
,
z
= 0.
A surface is bounded if it can be contained in a solid sphere, unbounded
otherwise. A bounded surface with no boundary is called closed (e.g. sphere).
Example.
The boundary of a hemisphere is a circle (drawn in red).
Definition
(Orientable surface)
.
At each point, there is a unit normal
n
that’s
unique up to a sign.
If we can find a consistent choice of
n
that varies smoothly across
S
, then
we say
S
is orientable, and the choice of sign of
n
is called the orientation of the
surface.
Most surfaces we encounter are orientable. For example, for a sphere, we can
declare that the normal should always point outwards. A notable example of a
non-orientable surface is the M¨obius strip (or Klein bottle).
For simple cases, we can describe the orientation as “inward” and “outward”.