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 .
Calculate the circulation of
$\text{F}(x,y)=\u27e8-\frac{y}{{x}^{2}+{y}^{2}},\frac{x}{{x}^{2}+{y}^{2}}\u27e9$ along a unit circle oriented counterclockwise.
Calculate the work done on a particle that traverses circle
C of radius 2 centered at the origin, oriented counterclockwise, by field
$\text{F}(x,y)=\u27e8\mathrm{-2},y\u27e9.$ Assume the particle starts its movement at
$(1,0).$
The work done by
F on the particle is the circulation of
F along
C :
${\oint}_{C}\text{F}\xb7\text{T}ds}.$ We use the parameterization
$\text{r}(t)=\u27e82\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\u27e9,0\le t\le 2\pi $ for
C . Then,
${r}^{\prime}(t)=\u27e8\mathrm{-2}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t,2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\u27e9$ and
$\text{F}\left(\text{r}(t)\right)=\u27e8\mathrm{-2},2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\u27e9.$ Therefore, the circulation of
F along
C is
Notice that the circulation of
F along
C is zero. Furthermore, notice that since
F is the gradient of
$f(x,y)=\mathrm{-2}x+\frac{{y}^{2}}{2},$F is conservative. We prove in a later section that under certain broad conditions, the circulation of a conservative vector field along a closed curve is zero.
Calculate the work done by field
$\text{F}(x,y)=\u27e82x,3y\u27e9$ on a particle that traverses the unit circle. Assume the particle begins its movement at
$(\mathrm{-1},0).$
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
${\int}_{C}ds}.$
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 $\int}_{C}f\left(x,y,z\right)ds={\displaystyle {\int}_{a}^{b}f\left(\text{r}(t)\right)\sqrt{{\left({x}^{\prime}(t)\right)}^{2}+{\left({y}^{\prime}(t)\right)}^{2}+{\left({z}^{\prime}(t)\right)}^{2}}dt$
Calculating a vector line integral ${\int}_{C}\text{F}\xb7ds}={\displaystyle {\int}_{C}\text{F}\xb7\text{T}ds}={\displaystyle {\int}_{a}^{b}\text{F}\left(\text{r}(t)\right)\xb7{{r}^{\prime}}^{}(t)dt$ or
${\int}_{C}Pdx+Qdy+Rdz}={\displaystyle {\int}_{a}^{b}\left(P(\text{r}(t))\frac{dx}{dt}+Q(\text{r}(t))\frac{dy}{dt}+R(\text{r}(t))\frac{dz}{dt}\right)dt$
True or False? Line integral
${\int}_{C}^{}f(x,y)ds$ is equal to a definite integral if
C is a smooth curve defined on
$\left[a,b\right]$ and if function
$f$ is continuous on some region that contains curve
C .
True or False? Vector functions
${\text{r}}_{1}=t\text{i}+{t}^{2}\text{j},$$0\le t\le 1,$ and
${\text{r}}_{2}=(1-t)\text{i}+{(1-t)}^{2}\text{j},$$0\le t\le 1,$ define the same oriented curve.
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Rafiq
Rafiq
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Mahi
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How this robot is carried to required site of body cell.?
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Rafiq
Rafiq
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Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?