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 .
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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
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Abdullahi
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DENG
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A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,