Determine the length of a particle’s path in space by using the arc-length function.
Explain the meaning of the curvature of a curve in space and state its formula.
Describe the meaning of the normal and binormal vectors of a curve in space.
In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. For example, suppose a vector-valued function describes the motion of a particle in space. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. Or, suppose that the vector-valued function describes a road we are building and we want to determine how sharply the road curves at a given point. This is described by the curvature of the function at that point. We explore each of these concepts in this section.
Arc length for vector functions
We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall
[link] , which states that the formula for the arc length of a curve defined by the parametric functions
$x=x\left(t\right),y=y\left(t\right),{t}_{1}\le t\le {t}_{2}$ is given by
In a similar fashion, if we define a smooth curve using a vector-valued function
$\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{j},$ where
$a\le t\le b,$ the arc length is given by the formula
In three dimensions, if the vector-valued function is described by
$\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ over the same interval
$a\le t\le b,$ the arc length is given by
Plane curve : Given a smooth curve
C defined by the function
$\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{j},$ where
t lies within the interval
$\left[a,b\right],$ the arc length of
C over the interval is
Space curve : Given a smooth curve
C defined by the function
$\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.2em}{0ex}}\text{k},$ where
t lies within the interval
$\left[a,b\right],$ the arc length of
C over the interval is
The two formulas are very similar; they differ only in the fact that a space curve has three component functions instead of two. Note that the formulas are defined for smooth curves: curves where the vector-valued function
$\text{r}(t)$ is differentiable with a non-zero derivative. The smoothness condition guarantees that the curve has no cusps (or corners) that could make the formula problematic.
Finding the arc length
Calculate the arc length for each of the following vector-valued functions:
Using
[link] ,
${r}^{\prime}\left(t\right)=\u27e8\text{cos}\phantom{\rule{0.1em}{0ex}}t-t\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t,\text{sin}\phantom{\rule{0.1em}{0ex}}t+t\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t,2\u27e9,$ so
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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.