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
$f\left(x+h\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,y+h\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)\text{:}$
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?