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 Calculus volume 3
 Vector calculus
 Stokes’ theorem
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
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Almas
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
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Professor
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
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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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