# 6.7 Stokes’ theorem

 Page 7 / 10

Calculate the curl of electric field E if the corresponding magnetic field is $\text{B}\left(t\right)=⟨tx,ty,-2tz⟩,0\le t<\infty .$

$\text{curl}\phantom{\rule{0.2em}{0ex}}\text{E}=⟨x,y,-2z⟩$

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
${\int }_{C}\text{F}·d\text{r}={\iint }_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S}$

For the following exercises, without using Stokes’ theorem, calculate directly both the flux of $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}$ over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

$\text{F}\left(x,y,z\right)={y}^{2}\text{i}+{z}^{2}\text{j}+{x}^{2}\text{k}\text{;}$ S is the first-octant portion of plane $x+y+z=1.$

$\text{F}\left(x,y,z\right)=z\text{i}+x\text{j}+y\text{k}\text{;}$ S is hemisphere $z={\left({a}^{2}-{x}^{2}-{y}^{2}\right)}^{1\text{/}2}.$

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=\pi {a}^{2}$

$\text{F}\left(x,y,z\right)={y}^{2}\text{i}+2x\text{j}+5\text{k}\text{;}$ S is hemisphere $z={\left(4-{x}^{2}-{y}^{2}\right)}^{1\text{/}2}.$

$\text{F}\left(x,y,z\right)=z\text{i}+2x\text{j}+3y\text{k}\text{;}$ S is upper hemisphere $z=\sqrt{9-{x}^{2}-{y}^{2}}.$

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=18\pi$

$\text{F}\left(x,y,z\right)=\left(x+2z\right)\text{i}+\left(y-x\right)\text{j}+\left(z-y\right)\text{k}\text{;}$ S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

$\text{F}\left(x,y,z\right)=2y\text{i}-6z\text{j}+3x\text{k}\text{;}$ S is a portion of paraboloid $z=4-{x}^{2}-{y}^{2}$ and is above the xy -plane.

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=-8\pi$

For the following exercises, use Stokes’ theorem to evaluate ${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS$ for the vector fields and surface.

$\text{F}\left(x,y,z\right)=xy\text{i}-z\text{j}$ 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.

$\text{F}\left(x,y,z\right)=xy\text{i}+{x}^{2}\text{j}+{z}^{2}\text{k}\text{;}$ and C is the intersection of paraboloid $z={x}^{2}+{y}^{2}$ and plane $z=y,$ and using the outward normal vector.

${\iint }_{S}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}\right)dS=0$

$\text{F}\left(x,y,z\right)=4y\text{i}+z\text{j}+2y\text{k}$ and C is the intersection of sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ with plane $z=0,$ and using the outward normal vector

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 . ${\int }_{C}\text{F}·d\text{S}=0$

[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.

[T] Use a CAS and Stokes’ theorem to approximate line integral $\underset{C}{\int }\left(3ydx+2zdy-5xdz\right),$ 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.

${\int }_{C}\text{F}·d\text{S}=-9.4248$

[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.

Use Stokes’ theorem to evaluate ${\iint }_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S},$ where $\text{F}\left(x,y,z\right)={e}^{xy}\text{cos}\phantom{\rule{0.2em}{0ex}}z\text{i}+{x}^{2}z\text{j}+xy\text{k},$ and S is half of sphere $x=\sqrt{1-{y}^{2}-{z}^{2}},$ oriented out toward the positive x -axis.

$\underset{S}{\iint }\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·d\text{S}=0$

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations? By Saylor Foundation By Janet Forrester By Danielle Stephens By John Gabrieli By By Dewey Compton By Yasser Ibrahim By OpenStax By OpenStax By Jonathan Long