# 22. Parametric Surfaces and Surface Integrals

## e. Vector Surface Integrals

For a parametric curve, the *important* vector is the tangent vector, \(\vec{v}\).

For a parametric surface, the *important* vector is the normal vector, \(\vec{N}\).

## Review: Arc Length Differentials and Line Integrals

Recall from the chapter on Line Integrals:

The scalar differential of arc length and
the vector differential of arc length are
\[
ds=|\vec{v}|\,dt
\qquad \text{and} \qquad
d\vec{s}=\vec{v}\,dt
\]
The length of the *important* vector, \(|\vec{v}|\), appears in the
scalar differential.

The *important* vector itself, \(\vec{v}\), appears in the vector
differential.

Then the scalar line integral is:
\[
\int_A^B f\,ds=\int_a^b f(\vec r(t))|\vec{v}|\,dt
\]
and the vector line integral is:
\[
\int_A^B \vec{F}\cdot\,d\vec{s}
=\int_a^b \vec{F}(\vec r(t))\cdot\vec{v}\,dt
\]
Notice the *dot* (\(\cdot\)) after the \(\vec{F}\) and the arrow
on \(d\vec{s}\). These remind us to use the vector \(\vec{v}\) so we can
compute a *dot product*.

## 1. Surface Area Differentials and Surface Integrals

Similarly:

The scalar differential of surface area and
the vector differential of surface area are:
\[
dS=|\vec{N}|\,du\,dv
\qquad \text{and} \qquad
d\vec{S}=\vec{N}\,du\,dv
\]
The length of the *important* vector, \(|\vec{N}|\), appears in the
scalar differential.

The *important* vector itself, \(\vec{N}\), appears in the vector
differential.

Then the scalar surface integral is:
\[
\iint_S f\,dS=\iint_S f(\vec R(u,v))|\vec{N}|\,du\,dv
\]
and the vector surface integral is:
\[
\iint_S \vec{F}\cdot\,d\vec{S}
=\iint_S \vec{F}(\vec R(u,v))\cdot\vec{N}\,du\,dv
\]
Limits need to be put on the surface integral to describe the boundary.

Notice the *dot* (\(\cdot\)) after the \(\vec{F}\) and the arrow
on \(d\vec{S}\). These reminds us to use the vector \(\vec{N}\) so we can
compute a *dot product*.

Notice the parallels between the differentials of arc length and surface area
and the parallels between the line and surface integrals.

Notice the lower case \(s\) for curves and the upper case \(S\) for surfaces.

For a parametric curve, the *important* vector is the tangent vector, \(\vec{v}\).

For a parametric surface, the *important* vector is the normal vector, \(\vec{N}\).

Notice the length of the important vector in the scalar integrals and the
dot product between vectors for the vector integrals.

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