Calculate the work done on a particle that traverses circle
C of radius 2 centered at the origin, oriented counterclockwise, by field
Assume the particle starts its movement at
The work done by
F on the particle is the circulation of
F along
C :
We use the parameterization
for
C . Then,
and
Therefore, the circulation of
F along
C is
The force field does zero work on the particle.
Notice that the circulation of
F along
C is zero. Furthermore, notice that since
F is the gradient of
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.
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
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
Calculating a vector line integral
or
Calculating flux
True or False? Line integral
is equal to a definite integral if
C is a smooth curve defined on
and if function
is continuous on some region that contains curve
C .
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form