Surface Integral - or Line Integral?

In summary, Joe calculates the flow of air through a loop which has straight lines connecting (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0). He uses the speed of the flow to find the volume of air flowing per second.
  • #1
bon
559
0

Homework Statement



Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0).

Homework Equations





The Attempt at a Solution



So I don't know if this is meant to be a line integral or surface one?

My feeling is that it should be a surface integral over that pentagonal surface..I.e. double integral of F.n dS..where F is (-1,-1,1)..

Firstly, is this right? Secondly, how do I use the fact that the speed of flow is 0.4? Thirdly, how do I find the normal to the plane?! Finally, how do i integrate over the surface of the pentagon in the double integral?!

Thanks!
 
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  • #2
Hi bon! :smile:

Hint: it's not a plane! :wink:

(draw a cube, and then trace the line around the cube)
 
  • #3
Ah okay thanks - so do i use divergence theorem to simplify the surface integral over the cube?

How do I use the speed of the flow?

THanks!
 
  • #4
It might be simplest in this case to just shade in those parts of the faces of the cube that the line crosses, and calculate the flow for each shaded face separately. :wink:
 
  • #5
Im trying to do a similar problem to this..

As bon says though, I can't see how you use the speed being 0.4m/s here?!

What I would do is dot (-1,-1,1) with n hat for and integrate over two surfaces separately - one with corners (1,1,0), (1,0,0) and (1,1,1)

and then over the other surface..

But how do you use the speed on the vector field?
 
  • #6
So summing those two integrals I get..2-1/2 = 1.5

But as you say bon, I'm not sure how to use the speed being 0.4m/s
 
  • #7
joe:) said:
Im trying to do a similar problem to this..

As bon says though, I can't see how you use the speed being 0.4m/s here?!

Hi joe:)! :smile:

volume per second = length x area per second = speed x area

or, more precisely, = ∫ velocity "dot" normal d(area) :wink:
 
  • #8
tiny-tim said:
Hi joe:)! :smile:

volume per second = length x area per second = speed x area

or, more precisely, = ∫ velocity "dot" normal d(area) :wink:

Very helpful. Thanks!

So I need to find a velocity vector in the direction (-1,-1,1) with magnitude 0.4?

So is it 0.4/root3 (-1,-1,1) then do I just dot this with the two normals for the two surfaces and carry out the two double integrals as I did..?

Was I right in getting 1.5? In which case i guess the actual answer should be 4/5root3 - 0.4/2root3 = root3/5? Correct?

THANK YOU :)
 
  • #9
Hi joe:)! :smile:

(have a square-root: √ :wink:)
joe:) said:
Very helpful. Thanks!

So I need to find a velocity vector in the direction (-1,-1,1) with magnitude 0.4?

So is it 0.4/root3 (-1,-1,1) then do I just dot this with the two normals for the two surfaces and carry out the two double integrals as I did..?

That's right! :smile:

(but I haven't checked your actual figures)
 
  • #10
Thanks tiny-tim :)
 

What is a surface integral?

A surface integral is a mathematical concept that calculates the total value of a function over a two-dimensional surface. It is commonly used in physics and engineering to calculate quantities such as mass, surface area, and electric flux.

How is a surface integral different from a line integral?

A surface integral is calculated over a two-dimensional surface, while a line integral is calculated over a one-dimensional curve. Additionally, the limits of integration for a surface integral are given as a region on a surface, while the limits for a line integral are given as points on a curve.

What is the significance of surface integrals in science?

Surface integrals are used in many fields of science, including physics, engineering, and mathematics. They allow for the calculation of various physical quantities, such as electric and magnetic fields, as well as the calculation of surface area and mass in geometry and calculus.

What are some common applications of surface integrals?

Surface integrals are commonly used in vector calculus to solve problems in electromagnetism, fluid mechanics, and heat transfer. They are also used in computer graphics to calculate lighting and shading on 3D surfaces.

How are surface integrals calculated?

The calculation of a surface integral involves breaking the surface into small, manageable pieces and summing the values of the function over each piece. This can be done using various methods, including double integrals and parametric equations.

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