Determine the directional derivative in a given direction for a function of two variables.
Determine the gradient vector of a given real-valued function.
Explain the significance of the gradient vector with regard to direction of change along a surface.
Use the gradient to find the tangent to a level curve of a given function.
Calculate directional derivatives and gradients in three dimensions.
In
Partial Derivatives we introduced the partial derivative. A function
$z=f\left(x,y\right)$ has two partial derivatives:
$\partial z\text{/}\partial x$ and
$\partial z\text{/}\partial y.$ These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). For example,
$\partial z\text{/}\partial x$ represents the slope of a tangent line passing through a given point on the surface defined by
$z=f\left(x,y\right),$ assuming the tangent line is parallel to the
x -axis. Similarly,
$\partial z\text{/}\partial y$ represents the slope of the tangent line parallel to the
$y\text{-axis.}$ Now we consider the possibility of a tangent line parallel to neither axis.
Directional derivatives
We start with the graph of a surface defined by the equation
$z=f\left(x,y\right).$ Given a point
$\left(a,b\right)$ in the domain of
$f,$ we choose a direction to travel from that point. We measure the direction using an angle
$\theta ,$ which is measured counterclockwise in the
x ,
y -plane, starting at zero from the positive
x -axis (
[link] ). The distance we travel is
$h$ and the direction we travel is given by the unit vector
$u=\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)i+\left(\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)j.$ Therefore, the
z -coordinate of the second point on the graph is given by
$z=f\left(a+h\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,b+h\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right).$
We can calculate the slope of the secant line by dividing the difference in
$z\text{-values}$ by the length of the line segment connecting the two points in the domain. The length of the line segment is
$h.$ Therefore, the slope of the secant line is
To find the slope of the tangent line in the same direction, we take the limit as
$h$ approaches zero.
Definition
Suppose
$z=f\left(x,y\right)$ is a function of two variables with a domain of
$D.$ Let
$\left(a,b\right)\in D$ and define
$\text{u}=\text{cos}\phantom{\rule{0.2em}{0ex}}\theta i+\text{sin}\phantom{\rule{0.2em}{0ex}}\theta j.$ Then the
directional derivative of
$f$ in the direction of
$u$ is given by
[link] provides a formal definition of the directional derivative that can be used in many cases to calculate a directional derivative.
Finding a directional derivative from the definition
Let
$\theta =\text{arccos}\left(3\text{/}5\right).$ Find the directional derivative
${D}_{u}f\left(x,y\right)$ of
$f\left(x,y\right)={x}^{2}-xy+3{y}^{2}$ in the direction of
$\text{u}=\left(\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)i+\left(\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)j.$ What is
${D}_{\text{u}}f\left(\mathrm{-1},2\right)?$
First of all, since
$\text{cos}\phantom{\rule{0.2em}{0ex}}\theta =3\text{/}5$ and
$\theta $ is acute, this implies
Using
$f\left(x,y\right)={x}^{2}-xy+3{y}^{2},$ we first calculate
$f\left(x+h\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,y+h\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)\text{:}$
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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.