<< Chapter < Page Chapter >> Page >

Work the previous example for surface S that is a sphere of radius 4 centered at the origin, oriented outward.

6.777 × 10 9

Got questions? Get instant answers now!

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

  • Divergence theorem
    E div F d V = S F · d S

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral S F · n d s for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

[T] F ( x , y , z ) = x i + y j + z k ; S is the surface of cube 0 x 1 , 0 y 1 , 0 < z 1 .

Got questions? Get instant answers now!

[T] F ( x , y , z ) = ( cos y z ) i + e x z j + 3 z 2 k ; S is the surface of hemisphere z = 4 x 2 y 2 together with disk x 2 + y 2 4 in the xy -plane.

S F · n d s = 75.3982

Got questions? Get instant answers now!

[T] F ( x , y , z ) = ( x 2 + y 2 x 2 ) i + x 2 y j + 3 z k ; S is the surface of the five faces of unit cube 0 x 1 , 0 y 1 , 0 < z 1 .

Got questions? Get instant answers now!

[T] F ( x , y , z ) = x i + y j + z k ; S is the surface of paraboloid z = x 2 + y 2 for 0 z 9 .

S F · n d s = 127.2345

Got questions? Get instant answers now!

[T] F ( x , y , z ) = x 2 i + y 2 j + z 2 k ; S is the surface of sphere x 2 + y 2 + z 2 = 4 .

Got questions? Get instant answers now!

[T] F ( x , y , z ) = x i + y j + ( z 2 1 ) k ; S is the surface of the solid bounded by cylinder x 2 + y 2 = 4 and planes z = 0 and z = 1 .

S F · n d s = 37.6991

Got questions? Get instant answers now!

[T] F ( x , y , z ) = x y 2 i + y z 2 j + x 2 z k ; S is the surface bounded above by sphere ρ = 2 and below by cone φ = π 4 in spherical coordinates. (Think of S as the surface of an “ice cream cone.”)

Got questions? Get instant answers now!

[T] F ( x , y , z ) = x 3 i + y 3 j + 3 a 2 z k (constant a > 0 ) ; S is the surface bounded by cylinder x 2 + y 2 = a 2 and planes z = 0 and z = 1 .

S F · n d s = 9 π a 4 2

Got questions? Get instant answers now!

[T] Surface integral S F · d S , where S is the solid bounded by paraboloid z = x 2 + y 2 and plane z = 4 , and F ( x , y , z ) = ( x + y 2 z 2 ) i + ( y + z 2 x 2 ) j + ( z + x 2 y 2 ) k

Got questions? Get instant answers now!

Use the divergence theorem to calculate surface integral S F · d S , where F ( x , y , z ) = ( e y 2 ) i + ( y + sin ( z 2 ) ) j + ( z 1 ) k and S is upper hemisphere x 2 + y 2 + z 2 = 1 , z 0 , oriented upward.

S F · d S = π 3

Got questions? Get instant answers now!

Use the divergence theorem to calculate surface integral S F · d S , where F ( x , y , z ) = x 4 i x 3 z 2 j + 4 x y 2 z k and S is the surface bounded by cylinder x 2 + y 2 = 1 and planes z = x + 2 and z = 0 .

Got questions? Get instant answers now!

Use the divergence theorem to calculate surface integral S F · d S when F ( x , y , z ) = x 2 z 3 i + 2 x y z 3 j + x z 4 k and S is the surface of the box with vertices ( ±1 , ±2 , ±3 ) .

S F · d S = 0

Got questions? Get instant answers now!

Use the divergence theorem to calculate surface integral S F · d S when F ( x , y , z ) = z tan −1 ( y 2 ) i + z 3 ln ( x 2 + 1 ) j + z k and S is a part of paraboloid x 2 + y 2 + z = 2 that lies above plane z = 1 and is oriented upward.

Got questions? Get instant answers now!

[T] Use a CAS and the divergence theorem to calculate flux S F · d S , where F ( x , y , z ) = ( x 3 + y 3 ) i + ( y 3 + z 3 ) j + ( z 3 + x 3 ) k and S is a sphere with center (0, 0) and radius 2.

S F · d S = 241.2743

Got questions? Get instant answers now!

Use the divergence theorem to compute the value of flux integral S F · d S , where F ( x , y , z ) = ( y 3 + 3 x ) i + ( x z + y ) j + [ z + x 4 cos ( x 2 y ) ] k and S is the area of the region bounded by x 2 + y 2 = 1 , x 0 , y 0 , and 0 z 1 .

A vector field in three dimensions, with focus on the area with x > 0, y>0, and z>0. A quarter of a cylinder is drawn with center on the z axis. The arrows have positive x, y, and z components; they point away from the origin.
Got questions? Get instant answers now!

Use the divergence theorem to compute flux integral S F · d S , where F ( x , y , z ) = y j z k and S consists of the union of paraboloid y = x 2 + z 2 , 0 y 1 , and disk x 2 + z 2 1 , y = 1 , oriented outward. What is the flux through just the paraboloid?

D F · d S = π

Got questions? Get instant answers now!

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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?
Radek Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask