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Calculate the curl of electric field E if the corresponding magnetic field is B ( t ) = t x , t y , −2 t z , 0 t < .

curl E = x , y , −2 z

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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
    C F · d r = S curl F · d S

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of curl F · N over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

F ( x , y , z ) = y 2 i + z 2 j + x 2 k ; S is the first-octant portion of plane x + y + z = 1 .

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F ( x , y , z ) = z i + x j + y k ; S is hemisphere z = ( a 2 x 2 y 2 ) 1 / 2 .

S ( curl F · N ) d S = π a 2

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F ( x , y , z ) = y 2 i + 2 x j + 5 k ; S is hemisphere z = ( 4 x 2 y 2 ) 1 / 2 .

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F ( x , y , z ) = z i + 2 x j + 3 y k ; S is upper hemisphere z = 9 x 2 y 2 .

S ( curl F · N ) d S = 18 π

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F ( x , y , z ) = ( x + 2 z ) i + ( y x ) j + ( z y ) k ; S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

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F ( x , y , z ) = 2 y i 6 z j + 3 x k ; S is a portion of paraboloid z = 4 x 2 y 2 and is above the xy -plane.

S ( curl F · N ) d S = −8 π

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For the following exercises, use Stokes’ theorem to evaluate S ( curl F · N ) d S for the vector fields and surface.

F ( x , y , z ) = x y i z j and S is the surface of the cube 0 x 1 , 0 y 1 , 0 z 1 , except for the face where z = 0 , and using the outward unit normal vector.

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F ( x , y , z ) = x y i + x 2 j + z 2 k ; and C is the intersection of paraboloid z = x 2 + y 2 and plane z = y , and using the outward normal vector.

S ( curl F · N ) d S = 0

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F ( x , y , z ) = 4 y i + z j + 2 y 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 C [ 2 x y 2 z d x + 2 x 2 y z d y + ( x 2 y 2 2 z ) d z ] , where C is the curve given by x = cos t , y = sin t , z = sin t , 0 t 2 π , traversed in the direction of increasing t .

A vector field in three dimensional space. The arrows are larger the further they are from the x, y plane. The arrows curve up from below the x, y plane and slightly above it. The rest tend to curve down and horizontally. An oval-shaped curve is drawn through the middle of the space.

C F · d S = 0

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[T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral C ( y d x + z d y + x d z ) , where C is the intersection of plane x + y = 2 and surface x 2 + y 2 + z 2 = 2 ( x + y ) , traversed counterclockwise viewed from the origin.

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[T] Use a CAS and Stokes’ theorem to approximate line integral C ( 3 y d x + 2 z d y 5 x d z ) , where C is the intersection of the xy -plane and hemisphere z = 1 x 2 y 2 , traversed counterclockwise viewed from the top—that is, from the positive z -axis toward the xy -plane.

C F · d S = −9.4248

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[T] Use a CAS and Stokes’ theorem to approximate line integral C [ ( 1 + y ) z d x + ( 1 + z ) x d y + ( 1 + x ) y d z ] , where C is a triangle with vertices ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , and ( 0 , 0 , 1 ) oriented counterclockwise.

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Use Stokes’ theorem to evaluate S curl F · d S , where F ( x , y , z ) = e x y cos z i + x 2 z j + x y k , and S is half of sphere x = 1 y 2 z 2 , oriented out toward the positive x -axis.

S curl F · d S = 0

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
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
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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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