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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Supported in part by NSF Grant #1123255