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Lessons for Multivariable Calculus

Mark Fitch

Section 5.8 Surface Vector Integrals

Section 5.2 presented how to calculate the affect of a vector field on a path. This section presents how to calculate the affect of a vector field on a surface.

Subsection 5.8.1 Illustration

Figure 5.8.1. Vector field passing through various surfaces

Activity 78.

The following problems illustrate how a field affects a surface.
(a)
Rank the three surfaces according to the effect of the field on them. It may help to think of the surfaces as sails and the field as wind.
(b)
Why is the effect greater on some surfaces?
(c)
Locate a point on the plane (red surface) where the affect of the field is maximum.
(d)
Explain the relationship of the field vector to the plane at that point.
(e)
Locate a point on the plane (red surface) where the affect of the field is not maximal.
(f)
Explain the relationship of the field vector to the plane at that point.
(g)
Explain how you could calculate the portion of the field vector that is maximally affecting the surface. This is the same as calculating the portion of the field vector that is in the same direction as a vector that would maximally affect the surface (like the previous questions).

Subsection 5.8.2 Calculation

For a continuous vector field \(F(x,y,z)\) and an parameterized oriented surface \(S(t,u)\) the surface integral of \(F\) over \(S\) is
\begin{equation*} \dint \vec{F} \cdot \vec{n} \; dS = \dint \vec{F}(S(t,u)) \cdot \left( \frac{\partial S}{\partial t} \times \frac{\delta S}{\delta u} \right) \; dA. \end{equation*}

Exercises 5.8.3 Exercises

Calculate the vector surface integral for each field over the specified surface.

1.

\(F(x,y,z)=(yz,xz,xy)\) over the cone \(S(\theta,h)=(h\cos\theta,h\sin\theta,h)\) for \(\theta \in [0,2\pi]\) and \(h \in [0,1].\)

2.

\(F(x,y,z)=(y+z,x+z,x+y)\) over the helicoid \(S(\theta,u)=(u\cos\theta,u\sin\theta,\theta)\) for \(\theta \in [0,\pi]\) and \(u \in [0,1].\)
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