6.2 Line integrals  (Page 10/20)

In [link] , what if we had oriented the unit circle clockwise? We denote the unit circle oriented clockwise by $\text{−}C.$ Then

Notice that the circulation is negative in this case. The reason for this is that the orientation of the curve flows against the direction of F .

Key concepts

• Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space.
• If C is a curve, then the length of C is ${\displaystyle {\int }_{C}ds}.$
• There are two kinds of line integral: scalar line integrals and vector line integrals. Scalar line integrals can be used to calculate the mass of a wire; vector line integrals can be used to calculate the work done on a particle traveling through a field.
• Scalar line integrals can be calculated using [link] ; vector line integrals can be calculated using [link] .
• Two key concepts expressed in terms of line integrals are flux and circulation. Flux measures the rate that a field crosses a given line; circulation measures the tendency of a field to move in the same direction as a given closed curve.

Key equations

• Calculating a scalar line integral
  ${\displaystyle {\int }_{C}f\left(x,y,z\right)ds={\displaystyle {\int }_a^{b}f\left(\text{r}(t)\right)\sqrt{{\left(x^{\prime }(t)\right)}^2+{\left(y^{\prime }(t)\right)}^2+{\left(z^{\prime }(t)\right)}^2}dt}}$
• Calculating a vector line integral
  ${\displaystyle {\int }_C\text{F}·ds}={\displaystyle {\int }_C\text{F}·\text{T}ds}={\displaystyle {\int }_a^b\text{F}\left(\text{r}(t)\right)·{r^{\prime }}^{}(t)dt}$
  or
  ${\displaystyle {\int }_{C}Pdx+Qdy+Rdz}={\displaystyle {\int }_a^b\left(P(\text{r}(t))\frac{dx}{dt}+Q(\text{r}(t))\frac{dy}{dt}+R(\text{r}(t))\frac{dz}{dt}\right)dt}$
• Calculating flux
  ${\displaystyle {\int }_C\text{F}·\frac{\text{n}(t)}{\Vert \text{n}(t)\Vert }}ds={\displaystyle {\int }_a^b\text{F}\left(\text{r}(t)\right)·\text{n}(t)dt}$