6.2 Line integrals  (Page 9/20)

We now give a formula for calculating the flux across a curve. This formula is analogous to the formula used to calculate a vector line integral (see [link] ).

Proof

The proof of [link] is similar to the proof of [link] . Before deriving the formula, note that $\Vert \text{n}(t)\Vert =\Vert ⟨y\prime (t),\text{−}x\prime (t)⟩\Vert =\sqrt{{\left(y\prime (t)\right)}^2+{\left(x\prime (t)\right)}^2}=\Vert r^{\prime }(t)\Vert .$ Therefore,

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Flux across a curve

Calculate the flux of $\text{F}=⟨2x,2y⟩$ across a unit circle oriented counterclockwise ( [link] ).

To compute the flux, we first need a parameterization of the unit circle. We can use the standard parameterization $\text{r}(t)=⟨\text{cos}t,\text{sin}t⟩,$ $0\le t\le 2\pi .$ The normal vector to a unit circle is $⟨\text{cos}t,\text{sin}t⟩.$ Therefore, the flux is

$\begin{array}{cc}\hfill {\displaystyle {\int }_C\text{F}·\text{N}}ds& ={\displaystyle {\int }_0^{2\pi }⟨2\text{cos}t,2\text{sin}t⟩·⟨\text{cos}t,\text{sin}t⟩}dt\hfill \\ & ={\displaystyle {\int }_0^{2\pi }\left(2{\text{cos}}^{2}t+2{\text{sin}}^{2}t\right)}dt=2{\displaystyle {\int }_0^{2\pi }\left({\text{cos}}^{2}t+{\text{sin}}^{2}t\right)}dt\hfill \\ & =2{\displaystyle {\int }_0^{2\pi }dt}=4\pi .\hfill \end{array}$

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Let $\text{F}(x,y)=⟨P(x,y),Q(x,y)⟩$ be a two-dimensional vector field. Recall that integral ${\displaystyle {\int }_C\text{F}·\text{T}}ds$ is sometimes written as ${\displaystyle {\int }_{C}Pdx+Qdy}.$ Analogously, flux ${\displaystyle {\int }_C\text{F}·\text{N}ds}$ is sometimes written in the notation ${\displaystyle {\int }_C\text{−}Qdx+Pdy,}$ because the unit normal vector N is perpendicular to the unit tangent T . Rotating the vector $d\text{r}=⟨dx,dy⟩$ by 90° results in vector $⟨dy,\text{−}dx⟩.$ Therefore, the line integral in [link] can be written as ${\displaystyle {\int }_C\mathrm{-2}ydx+2xdy}.$

Now that we have defined flux, we can turn our attention to circulation. The line integral of vector field F along an oriented closed curve is called the circulation    of F along C . Circulation line integrals have their own notation: ${\displaystyle {\oint }_C\text{F}·\text{T}ds}.$ The circle on the integral symbol denotes that C is “circular” in that it has no endpoints. [link] shows a calculation of circulation.

To see where the term circulation comes from and what it measures, let v represent the velocity field of a fluid and let C be an oriented closed curve. At a particular point P , the closer the direction of v ( P ) is to the direction of T ( P ), the larger the value of the dot product $\text{v}(P)·\text{T}(P).$ The maximum value of $\text{v}(P)·\text{T}(P)$ occurs when the two vectors are pointing in the exact same direction; the minimum value of $\text{v}(P)·\text{T}(P)$ occurs when the two vectors are pointing in opposite directions. Thus, the value of the circulation ${\displaystyle {\oint }_C\text{v}·\text{T}ds}$ measures the tendency of the fluid to move in the direction of C .