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Key concepts
- The divergence theorem relates a surface integral across closed surface
S to a triple integral over the solid enclosed by
S . The divergence theorem is a higher dimensional version of the flux form of Green’s theorem, and is therefore a higher dimensional version of the Fundamental Theorem of Calculus.
- The divergence theorem can be used to transform a difficult flux integral into an easier triple integral and vice versa.
- The divergence theorem can be used to derive Gauss’ law, a fundamental law in electrostatics.
Key equations
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Divergence theorem
For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral
for the given choice of
F and the boundary surface
S. For each closed surface, assume
N is the outward unit normal vector.
[T]
S is the surface of hemisphere
together with disk
in the
xy -plane.
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[T]
S is the surface bounded above by sphere
and below by cone
in spherical coordinates. (Think of
S as the surface of an “ice cream cone.”)
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[T]
S is the surface bounded by cylinder
and planes
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[T] Surface integral
where
S is the solid bounded by paraboloid
and plane
and
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Use the divergence theorem to calculate surface integral
where
and
S is upper hemisphere
oriented upward.
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Use the divergence theorem to calculate surface integral
where
and S is the surface bounded by cylinder
and planes
and
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Use the divergence theorem to calculate surface integral
when
and
S is the surface of the box with vertices
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Use the divergence theorem to calculate surface integral
when
and
S is a part of paraboloid
that lies above plane
and is oriented upward.
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[T] Use a CAS and the divergence theorem to calculate flux
where
and
S is a sphere with center (0, 0) and radius 2.
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Use the divergence theorem to compute the value of flux integral
where
and
S is the area of the region bounded by
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Use the divergence theorem to compute flux integral
where
and
S consists of the union of paraboloid
and disk
oriented outward. What is the flux through just the paraboloid?
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Questions & Answers
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
learn
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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 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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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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?
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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