Calculus Small Changes, Big Ideas

Section 19.2 Work, circulation, and flux

• If \(\bv F\) is a force field that acts on an object moving along a path \(C\) parametrized by \(\bv r(t)\text{,}\) \(a\le t\le b\text{,}\) then the line integral of \(\bv F\) along \(C\) is

  \begin{equation*} \int_{C}\bv F\cdot\dd{\bv{r}}=\int_{a}^{b}\bv F(\bv r(t))\cdot\bv r'(t)\dd{t}\text{.} \end{equation*}

  This is the work done by \(\bv F\) on the object as it moves along \(C\text{.}\)

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• We can rewrite this as

  \begin{equation*} \int_{a}^{b}\qty(\bv F(\bv r(t))\cdot\frac{\bv r'(t)}{\norm{\bv r'(t)}})\norm{\bv r'(t)}\dd{t}=\int_{a}^{b}\bv F\cdot\uv{t}\dd{s} \end{equation*}

  That is, we’re really integrating the component of \(\bv F\) in the tangent direction of the curve.

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• If \(\bv F(x,y)=(P(x,y),Q(x,y))\text{,}\) we have:

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  \begin{align*} \bv{F}(x,y)\cdot\dd{\bv{r}} \amp = \int_a^b \qty(P(x(t),y(t))x'(t)+Q(x(t),y(t))y'(t))\dd{t}\\ \amp = \int_a^b P(x(t),y(t),z(t))\dd{t}+\int_a^b Q(x(t),y(t),z(t))\dd{t}\\ \amp = \int_C P(x,y)\dd{x}+\int_C Q(x,y)\dd{y} \end{align*}
  This is often abbreviated as \(\int_{C} P(x,y)\dd{x}+Q(x,y)\dd{y}\text{.}\)

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• Suppose \(C\) is a closed curve.

  • If we compute

    \begin{equation*} \oint_{C}\bv F\cdot\uv{n}\dd{s}\text{,} \end{equation*}

    where \(\uv n\) is the outward-pointing unit normal, we call this the flux of \(\bv F\) through \(C\text{.}\) The notation means we’re computing an integral of a closed curve. We can think of this as the amount of fluid that passes through the curve per unit time.

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  • If we compute

    \begin{equation*} \oint_{C}\bv F\cdot\uv{t}\dd{s}\text{,} \end{equation*}

    we call this the circulation of \(\bv F\) along \(C\text{.}\) We can think of this as the tendency of the fluid to flow along \(C\text{.}\)

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  • If \(C\) is parametrized counterclockwise, we say it positively oriented.

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• Work and circulation can be extended to 3D. Flux will have to wait.

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