Finding total outward flux through a pyramid?

In summary, the problem involves finding the surface integral of F=xi-yj+zk over a pyramid with five faces. Using the divergence theorem, the answer is found to be 1/3. The normal vectors for the two angled faces are i+k and j+k, and the limits of integration can be determined by drawing a diagonal in the xy plane and expressing the surface as a vector function R(x,y). The non-angled faces have a flux of zero.
  • #1
Infernorage
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Homework Statement
Let F=xi-yj+zk and the pyramid be as shown below. Find [tex]\iint\limits \,[/tex][tex]F\bullet dS[/tex] as a surface integral over all five faces.
pyramid.jpg


The attempt at a solution
I first solved it using the divergence theorem. The divergence was just 1, so the answer should just be the volume of the pyramid, which is 1/3, right?

I think that the normal vectors of the two angled faces are [tex]i+k[/tex] and [tex]j+k[/tex], but I don't understand how to find the dS or the limits of integration since no real equation was given for the surface. Tried doing the surface integrals of the two angled faces assuming that the two angled faces were the intersection of the surfaces [tex]z=1-x[/tex] and [tex]z=1-y[/tex], but my answer is not coming out to 1/3. The flux through the non-angled faces (left, back, and bottom) should just be zero, right? I don't know whether I am just not seeing it or what, but can you guys help me out? Thanks in advance.
 
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  • #2
Infernorage said:
Homework Statement
Let F=xi-yj+zk and the pyramid be as shown below. Find [tex]\iint\limits \,[/tex][tex]F\bullet dS[/tex] as a surface integral over all five faces.
pyramid.jpg


The attempt at a solution
I first solved it using the divergence theorem. The divergence was just 1, so the answer should just be the volume of the pyramid, which is 1/3, right?

Yes, that looks correct.

I think that the normal vectors of the two angled faces are [tex]i+k[/tex] and [tex]j+k[/tex], but I don't understand how to find the dS or the limits of integration since no real equation was given for the surface.

Tried doing the surface integrals of the two angled faces assuming that the two angled faces were the intersection of the surfaces [tex]z=1-x[/tex] and [tex]z=1-y[/tex], but my answer is not coming out to 1/3.

In the xy plane draw the diagonal from (0,0,0) to (1,1,0). That is the line y = x in the xy plane and the two triangles it forms lie directly under the two slanted faces. So if you express the integrals in terms of x and y, you can read the limits in the xy plane.

As for calculating the vector dS, if you express the surface as a vector function R(x,y), using x and y as the independent variables you can use the formula

[tex]\int\int_S \vec F \cdot d\vec S = \pm\int\int_R \vec F \cdot \vec R_x \times\vec R_y\,dxdy[/tex]

where the sign is chosen so the direction of Rx X Ry agrees with the orientiation of the surface.

For example, the front slanted plane x + z = 1 can be written:

R(x,y) = < x, y, 1-x >

Just use the formula above and express everything in terms of x. Read the limits from the xy plane. Similarly for the other slanted plane.

The flux through the non-angled faces (left, back, and bottom) should just be zero, right?

That's right. I presume you know why, and if this is a homework problem, you should explain why.
 

Related to Finding total outward flux through a pyramid?

1. What is total outward flux?

Total outward flux is the measure of the flow of a vector field through a given surface. It represents the amount of a vector field that is passing through the surface in the outward direction.

2. How is total outward flux calculated?

Total outward flux can be calculated using the surface integral of the dot product of the vector field and the surface's outward unit normal vector. This is also known as the flux density.

3. What is a pyramid in terms of flux calculation?

In terms of flux calculation, a pyramid is a three-dimensional shape with a polygonal base and triangular faces that meet at a common point. It can be used to represent a surface for calculating total outward flux.

4. What is the importance of finding total outward flux through a pyramid?

Finding total outward flux through a pyramid is important in many real-world applications, such as analyzing fluid flow through a pyramid-shaped object or calculating the amount of electric charge passing through a pyramid-shaped surface.

5. Are there any special considerations when finding total outward flux through a pyramid?

Yes, when finding total outward flux through a pyramid, it is important to make sure that the surface is closed and that the flux density is constant over the entire surface. Additionally, the direction of the flux must be consistent with the orientation of the surface's normal vector.

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