which gives the relationship between the arc length
s and the parameter
t as
$s=4t;$ so,
$t=s\text{/}4.$ Next we replace the variable
t in the original function
$\text{r}\left(t\right)=4\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.2em}{0ex}}\text{i}+4\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.2em}{0ex}}\text{j}$ with the expression
$s\text{/}4$ to obtain
This is the arc-length parameterization of
$\text{r}\left(t\right).$ Since the original restriction on
t was given by
$t\ge 0,$ the restriction on
s becomes
$s\text{/}4\ge 0,$ or
$s\ge 0.$
Therefore, the relationship between the arc length
s and the parameter
t is
$s=3t-9,$ so
$t=\frac{s}{3}+3.$ Substituting this into the original function
$\text{r}\left(t\right)=\u27e8t+3,\phantom{\rule{0.2em}{0ex}}2t-4,2t\u27e9$ yields
This is an arc-length parameterization of
$\text{r}\left(t\right).$ The original restriction on the parameter
$t$ was
$t\ge 3,$ so the restriction on
s is
$\left(s\text{/}3\right)+3\ge 3,$ or
$s\ge 0.$
Then, use the relationship between the arc length and the parameter
t to find an arc-length parameterization of
$\text{r}\left(t\right).$
$s=5t,$ or
$t=s\text{/}5.$ Substituting this into
$\text{r}\left(t\right)=\u27e83\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t,3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t,4t\u27e9$ gives
An important topic related to arc length is curvature. The concept of curvature provides a way to measure how sharply a smooth curve turns. A circle has constant curvature. The smaller the radius of the circle, the greater the curvature.
Think of driving down a road. Suppose the road lies on an arc of a large circle. In this case you would barely have to turn the wheel to stay on the road. Now suppose the radius is smaller. In this case you would need to turn more sharply to stay on the road. In the case of a curve other than a circle, it is often useful first to inscribe a circle to the curve at a given point so that it is tangent to the curve at that point and “hugs” the curve as closely as possible in a neighborhood of the point (
[link] ). The curvature of the graph at that point is then defined to be the same as the curvature of the inscribed circle.
Definition
Let
C be a smooth curve in the plane or in space given by
$\text{r}\left(s\right),$ where
$s$ is the arc-length parameter. The
curvature$\kappa $ at
s is
Visit this
website for more information about the curvature of a space curve.
The formula in the definition of curvature is very useful in terms of calculation. In particular, recall that
$\text{T}\left(t\right)$ represents the unit tangent vector to a given vector-valued function
$\text{r}\left(t\right),$ and the formula for
$\text{T}\left(t\right)$ is
$\text{T}\left(t\right)=\frac{{r}^{\prime}\left(t\right)}{\Vert {r}^{\prime}\left(t\right)\Vert}.$ To use the formula for curvature, it is first necessary to express
$\text{r}\left(t\right)$ in terms of the arc-length parameter
s , then find the unit tangent vector
$\text{T}\left(s\right)$ for the function
$\text{r}\left(s\right),$ then take the derivative of
$\text{T}\left(s\right)$ with respect to
s. This is a tedious process. Fortunately, there are equivalent formulas for curvature.
Questions & Answers
can someone help me with some logarithmic and exponential equations.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.