<< Chapter < Page Chapter >> Page >

Calculate the mass of a spring in the shape of a helix parameterized by r ( t ) = cos t , sin t , t , 0 t 6 π , with a density function given by ρ ( x , y , z ) = x + y + z kg/m.

18 2 π 2 kg

Got questions? Get instant answers now!

When we first defined vector line integrals, we used the concept of work to motivate the definition. Therefore, it is not surprising that calculating the work done by a vector field representing a force is a standard use of vector line integrals. Recall that if an object moves along curve C in force field F , then the work required to move the object is given by C F · d r .

Calculating work

How much work is required to move an object in vector force field F = y z , x y , x z along path r ( t ) = t 2 , t , t 4 , 0 t 1 ? See [link] .

Let C denote the given path. We need to find the value of C F · d r . To do this, we use [link] :

C F · d r = 0 1 ( t 5 , t 3 , t 6 · 2 t , 1 , 4 t 3 ) d t = 0 1 ( 2 t 6 + t 3 + 4 t 9 ) d t = [ 2 t 7 7 + t 4 4 + 2 t 10 5 ] t = 0 t = 1 = 131 140 .
A three dimensional diagram of the curve and vector field for the example. The curve is an increasing concave up curve starting close to the origin and above the x axis. As the curve goes left above the (x,y) plane, the height also increases. The arrows in the vector field get longer as the z component becomes larger.
The curve and vector field for [link] .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Flux and circulation

We close this section by discussing two key concepts related to line integrals: flux across a plane curve and circulation along a plane curve. Flux is used in applications to calculate fluid flow across a curve, and the concept of circulation is important for characterizing conservative gradient fields in terms of line integrals. Both these concepts are used heavily throughout the rest of this chapter. The idea of flux is especially important for Green’s theorem, and in higher dimensions for Stokes’ theorem and the divergence theorem.

Let C be a plane curve and let F be a vector field in the plane. Imagine C is a membrane across which fluid flows, but C does not impede the flow of the fluid. In other words, C is an idealized membrane invisible to the fluid. Suppose F represents the velocity field of the fluid. How could we quantify the rate at which the fluid is crossing C ?

Recall that the line integral of F along C is C F · T d s —in other words, the line integral is the dot product of the vector field with the unit tangential vector with respect to arc length. If we replace the unit tangential vector with unit normal vector N ( t ) and instead compute integral C F · N d s , we determine the flux across C . To be precise, the definition of integral C F · N d s is the same as integral C F · T d s , except the T in the Riemann sum is replaced with N . Therefore, the flux across C is defined as

C F · N d s = lim n i = 1 n F ( P i * ) · N ( P i * ) Δ s i ,

where P i * and Δ s i are defined as they were for integral C F · T d s . Therefore, a flux integral is an integral that is perpendicular to a vector line integral, because N and T are perpendicular vectors.

If F is a velocity field of a fluid and C is a curve that represents a membrane, then the flux of F across C is the quantity of fluid flowing across C per unit time, or the rate of flow.

More formally, let C be a plane curve parameterized by r ( t ) = x ( t ) , y ( t ) , a t b . Let n ( t ) = y ( t ) , x ( t ) be the vector that is normal to C at the endpoint of r ( t ) and points to the right as we traverse C in the positive direction ( [link] ). Then, N ( t ) = n ( t ) n ( t ) is the unit normal vector to C at the endpoint of r ( t ) that points to the right as we traverse C .


The flux    of F across C is line integral C F · n ( t ) n ( t ) d s .

Questions & Answers

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.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 8

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?