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Find the work done by vector field F ( x , y , z ) = x i + 3 x y j ( x + z ) k on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).

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How much work is required to move an object in vector field F ( x , y ) = y i + 3 x j along the upper part of ellipse x 2 4 + y 2 = 1 from (2, 0) to ( −2 , 0 ) ?

W = 2 π

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A vector field is given by F ( x , y ) = ( 2 x + 3 y ) i + ( 3 x + 2 y ) j . Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

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Evaluate the line integral of scalar function x y along parabolic path y = x 2 connecting the origin to point (1, 1).

C F · d r = 25 5 + 1 120

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Find C y 2 d x + ( x y x 2 ) d y along C : y = 3 x from (0, 0) to (1, 3).

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Find C y 2 d x + ( x y x 2 ) d y along C : y 2 = 9 x from (0, 0) to (1, 3).

C y 2 d x + ( x y x 2 ) d y = 6.15

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For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate F ( x , y , z ) = x 2 z i + 6 y j + y z 2 k , where C is represented by r ( t ) = t i + t 2 j + ln t k , 1 t 3 .

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[T] Evaluate line integral γ x e y d s where, γ is the arc of curve x = e y from ( 1 , 0 ) to ( e , 1 ) .

γ x e y d s 7.157

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[T] Evaluate the integral γ x y 2 d s , where γ is a triangle with vertices (0, 1, 2), (1, 0, 3), and ( 0 , −1 , 0 ) .

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[T] Evaluate line integral γ ( y 2 x y ) d x , where γ is curve y = ln x from (1, 0) toward ( e , 1 ) .

γ ( y 2 x y ) d x −1.379

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[T] Evaluate line integral γ x y 4 d s , where γ is the right half of circle x 2 + y 2 = 16 .

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[T] Evaluate C F · d r , where F ( x , y , z ) = x 2 y i + ( x z ) j + x y z k and

C : r ( t ) = t i + t 2 j + 2 k , 0 t 1 .

C F · d r −1.133

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Evaluate C F · d r , where F ( x , y ) = 2 x sin ( y ) i + ( x 2 cos ( y ) 3 y 2 ) j and

C is any path from ( −1 , 0 ) to (5, 1).

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Find the line integral of F ( x , y , z ) = 12 x 2 i 5 x y j + x z k over path C defined by y = x 2 , z = x 3 from point (0, 0, 0) to point (2, 4, 8).

C F · d r 2 2 . 8 5 7

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Find the line integral of C ( 1 + x 2 y ) d s , where C is ellipse r ( t ) = 2 cos t i + 3 sin t j from 0 t π .

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For the following exercises, find the flux.

Compute the flux of F = x 2 i + y j across a line segment from (0, 0) to (1, 2).

flux = 1 3

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Let F = 5 i and let C be curve y = 0 , 0 x 4 . Find the flux across C .

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Let F = 5 j and let C be curve y = 0 , 0 x 4 . Find the flux across C .

flux = −20

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Let F = y i + x j and let C : r ( t ) = cos t i + sin t j ( 0 t 2 π ) . Calculate the flux across C .

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Let F = ( x 2 + y 3 ) i + ( 2 x y ) j . Calculate flux F orientated counterclockwise across curve C : x 2 + y 2 = 9 .

flux = 0

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Find the line integral of C z 2 d x + y d y + 2 y d z , where C consists of two parts: C 1 and C 2 . C 1 is the intersection of cylinder x 2 + y 2 = 16 and plane z = 3 from (0, 4, 3) to ( −4 , 0 , 3 ) . C 2 is a line segment from ( −4 , 0 , 3 ) to (0, 1, 5).

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A spring is made of a thin wire twisted into the shape of a circular helix x = 2 cos t , y = 2 sin t , z = t . Find the mass of two turns of the spring if the wire has constant mass density.

m = 4 π ρ 5

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A thin wire is bent into the shape of a semicircle of radius a . If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

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An object moves in force field F ( x , y , z ) = y 2 i + 2 ( x + 1 ) y j counterclockwise from point (2, 0) along elliptical path x 2 + 4 y 2 = 4 to ( −2 , 0 ) , and back to point (2, 0) along the x -axis. How much work is done by the force field on the object?

W = 0

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Find the work done when an object moves in force field F ( x , y , z ) = 2 x i ( x + z ) j + ( y x ) k along the path given by r ( t ) = t 2 i + ( t 2 t ) j + 3 k , 0 t 1 .

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If an inverse force field F is given by F ( x , y , z ) = k r 3 r , where k is a constant, find the work done by F as its point of application moves along the x -axis from A ( 1 , 0 , 0 ) to B ( 2 , 0 , 0 ) .

W = k 2

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David and Sandra plan to evaluate line integral C F · d r along a path in the xy -plane from (0, 0) to (1, 1). The force field is F ( x , y ) = ( x + 2 y ) i + ( x + y 2 ) j . David chooses the path that runs along the x -axis from (0, 0) to (1, 0) and then runs along the vertical line x = 1 from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line y = x from (0, 0) to (1, 1). Whose line integral is larger and by how much?

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Questions & Answers

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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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