Calculus Small Changes, Big Ideas

Section 19.4 Surface integrals

• Just as curves can be written as functions of a parameter \(t\text{,}\) surfaces can be written as functions of two parameters, often called \(u\) and \(v\text{.}\)

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• A few examples:

  • Sphere of radius \(\rho\text{:}\)

    \begin{equation*} \bv r(\theta,\phi)=(x,y,z)=(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi) \end{equation*}

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  • Helicoid:

    \begin{equation*} \bv r(u,v)=(x,y,z)=(v\cos u,v\sin u,au) \end{equation*}

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  • Surface of revolution about \(x\)-axis:

    \begin{equation*} \bv r(u,v)=(x,y,z)=(u,f(u)\cos v,f(u)\sin v) \end{equation*}

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  For all of these, \((u,v)\) are points of some domain \(D\text{.}\)

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• Surface area element:

  \begin{equation*} \dd{S}=\norm{\pdv{\bv{r}}{u}\times\pdv{\bv{r}}{v}}\dd{A} \end{equation*}

  Here, \(\dd{A}=\dd{u}\dd{v}\text{.}\)

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• Surface area:

  \begin{equation*} S=\iint_{S}\dd{S}=\iint_{D}\norm{\pdv{\bv{r}}{u}\times\pdv{\bv{r}}{v}}\dd{A} \end{equation*}

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• Similarly to with line integrals, we can extend this to make the surface integral of a function over that surface:

  \begin{equation*} S=\iint_{S}f\dd{S}=\iint_{D}f(\bv r(u,v))\norm{\pdv{\bv{r}}{u}\times\pdv{\bv{r}}{v}}\dd{A} \end{equation*}

  Notice this is just the next level up of what we already do with line integrals.

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• If \(f\) is density, then we can compute mass, center of mass, moment of inertia, etc.

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• Again, if \(\bv r(u,v)=(x,y,z)\text{,}\) and the Jacobian is

  \begin{equation*} \bv J=\begin{bmatrix}\DS\pdv{x}{u} \amp \DS\pdv{x}{v}\\[2 ex] \DS\pdv{y}{u} \amp \DS\pdv{y}{v}\\[2 ex] \DS\pdv{z}{u} \amp \DS\pdv{z}{v}\end{bmatrix}\text{,} \end{equation*}

  then we end up with \(\sqrt{\abs{\det(\bv J^T\bv J)}}=\norm{\DS\pdv{\bv{r}}{u}\times\dv{\bv{r}}{v}}\text{.}\)

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