



Calculate the curl of electric field
E if the corresponding magnetic field is
$\text{B}(t)=\u27e8tx,ty,\mathrm{2}tz\u27e9,0\le t<\infty .$
$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{E}=\u27e8x,y,\mathrm{2}z\u27e9$
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Notice that the curl of the electric field does not change over time, although the magnetic field does change over time.
Key concepts
 Stokes’ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes’ theorem is a higher dimensional version of Green’s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.
 Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.
 Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary
C .
 Faraday’s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes’ theorem can be used to derive Faraday’s law.
Key equations

Stokes’ theorem
$\int}_{C}\text{F}\xb7d\text{r}}={\displaystyle {\iint}_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7d\text{S$
For the following exercises, without using Stokes’ theorem, calculate directly both the flux of
$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N}$ over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.
$\text{F}(x,y,z)=z\text{i}+x\text{j}+y\text{k}\text{;}$
S is hemisphere
$z={\left({a}^{2}{x}^{2}{y}^{2}\right)}^{1\text{/}2}.$
${\iint}_{S}(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N})dS}=\pi {a}^{2$
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$\text{F}(x,y,z)=z\text{i}+2x\text{j}+3y\text{k}\text{;}$
S is upper hemisphere
$z=\sqrt{9{x}^{2}{y}^{2}}.$
${\iint}_{S}(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N})dS}=18\pi $
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$\text{F}(x,y,z)=\left(x+2z\right)\text{i}+\left(yx\right)\text{j}+\left(zy\right)\text{k}\text{;}$
S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).
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$\text{F}(x,y,z)=2y\text{i}6z\text{j}+3x\text{k}\text{;}$
S is a portion of paraboloid
$z=4{x}^{2}{y}^{2}$ and is above the
xy plane.
${\iint}_{S}(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N})dS}=\mathrm{8}\pi $
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For the following exercises, use Stokes’ theorem to evaluate
${\iint}_{S}(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N})dS$ for the vector fields and surface.
$\text{F}(x,y,z)=xy\text{i}z\text{j}$ and
S is the surface of the cube
$0\le x\le 1,0\le y\le 1,0\le z\le 1,$ except for the face where
$z=0,$ and using the outward unit normal vector.
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$\text{F}(x,y,z)=xy\text{i}+{x}^{2}\text{j}+{z}^{2}\text{k}\text{;}$ and
C is the intersection of paraboloid
$z={x}^{2}+{y}^{2}$ and plane
$z=y,$ and using the outward normal vector.
${\iint}_{S}(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7\text{N})dS}=0$
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$\text{F}(x,y,z)=4y\text{i}+z\text{j}+2y\text{k}$ and
C is the intersection of sphere
${x}^{2}+{y}^{2}+{z}^{2}=4$ with plane
$z=0,$ and using the outward normal vector
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Use Stokes’ theorem to evaluate
$\underset{C}{\int}\left[2x{y}^{2}zdx+2{x}^{2}yzdy+\left({x}^{2}{y}^{2}2z\right)dz\right]},$ where
C is the curve given by
$x=\text{cos}\phantom{\rule{0.2em}{0ex}}t,y=\text{sin}\phantom{\rule{0.2em}{0ex}}t,z=\text{sin}\phantom{\rule{0.2em}{0ex}}t,0\le t\le 2\pi ,$ traversed in the direction of increasing
t .
${\int}_{C}\text{F}\xb7d\text{S}=0$
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[T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral
$\underset{C}{\int}\left(ydx+zdy+xdz\right)},$ where
C is the intersection of plane
$x+y=2$ and surface
${x}^{2}+{y}^{2}+{z}^{2}=2\left(x+y\right),$ traversed counterclockwise viewed from the origin.
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[T] Use a CAS and Stokes’ theorem to approximate line integral
$\underset{C}{\int}\left(3ydx+2zdy5xdz\right)},$ where
C is the intersection of the
xy plane and hemisphere
$z=\sqrt{1{x}^{2}{y}^{2}},$ traversed counterclockwise viewed from the top—that is, from the positive
z axis toward the
xy plane.
$\int}_{C}\text{F}\xb7d\text{S}=\mathrm{9.4248$
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[T] Use a CAS and Stokes’ theorem to approximate line integral
$\underset{C}{\int}\left[\left(1+y\right)zdx+\left(1+z\right)xdy+\left(1+x\right)ydz\right]},$ where
C is a triangle with vertices
$\left(1,0,0\right),$
$\left(0,1,0\right),$ and
$\left(0,0,1\right)$ oriented counterclockwise.
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Use Stokes’ theorem to evaluate
${\iint}_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7d\text{S}},$ where
$\text{F}(x,y,z)={e}^{xy}\text{cos}\phantom{\rule{0.2em}{0ex}}z\text{i}+{x}^{2}z\text{j}+xy\text{k},$ and
S is half of sphere
$x=\sqrt{1{y}^{2}{z}^{2}},$ oriented out toward the positive
x axis.
$\underset{S}{\iint}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7d\text{S}=0$
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Questions & Answers
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is system testing?
AMJAD
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field .
1Electronicsmanufacturad IC ,RAM,MRAM,solar panel etc
2Helth and MedicalNanomedicine,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
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:
OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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