Flux through a surface Question

In summary, the conversation discusses finding the flux of a vector field through a surface using different equations. One solution suggests parameterizing the surface in polar coordinates and using a single path integral, while another suggests using Cartesian coordinates and a double integral. The conversation also mentions incorrect solutions and suggests using a normal vector of k. The final solution calculates an answer of -238pi using polar coordinates.
  • #1
leext101
5
0

Homework Statement


Let S be the part of the surface z=49-(x2+y2)2 above the xy-plane, oriented upward.

Let vector field F= (yz) i +(xz) j + (-17+xy) k

Compute the flux of F through S.


Homework Equations


Flux through surface equation ∫s F(x,y,f(x,y)) dot product (-fx i-fy j + k) dxdy


The Attempt at a Solution


I used the equation to find flux through a surface plugging in F(x,y,(49-(x2+y2)2) for the vector field, I took the dot product. I believe the limits are -sqrt(7)≤x≤sqrt(7) and -sqrt(7)≤y≤sqrt(7). The answer I integrated out was -476 which was incorrect.

I appreciate your time and help!
 
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  • #2
Try parameterizing the surface in polar coordinates and then use

∫∫F.n ds = ∫∫F |rrxrθ| dA
 
  • #3
rock.freak667 said:
Try parameterizing the surface in polar coordinates and then use

∫∫F.n ds = ∫∫F |rrxrθ| dA
The first of these should be a single path integral, not a double integral, shouldn't it? Also I am puzzled by your "F.n". I would have used [itex]\vec{F}\cdot d\vec{s}[/itex] where "[itex]d\vec{s}[/itex]" is the vector tangent to the curve, not normal to it, with length ds.
 
  • #4
So if I do use polar coordinates would the limits be 0≤r≤sqrt(7), 0≤θ≤2∏ and a normal vector of n= k?
 
  • #5
using polar coordinates I calculated an answer -238pi
 
  • #6
If I stick with cartesian coordinates would the limits be -sqrt(7)≤y≤sqrt(7) and
-sqrt(7)≤x≤sqrt≤(7)?

Thanks everyone for posting
 

Related to Flux through a surface Question

1. What is flux through a surface?

Flux through a surface is a measure of the flow of a physical quantity, such as energy or particles, through a particular surface. It is a vector quantity that takes into account both the magnitude and direction of the flow.

2. How is flux through a surface calculated?

To calculate flux through a surface, you need to first determine the normal vector to the surface, then find the dot product between the vector and the flow vector of the physical quantity. This will give you the amount of flux passing through the surface in a certain direction.

3. What is the unit of measurement for flux through a surface?

The unit of measurement for flux through a surface depends on the physical quantity being measured. For example, if measuring energy flux, the unit would be watts per square meter. If measuring particle flux, the unit would be particles per square meter per second.

4. How is flux through a surface affected by the size of the surface?

The size of the surface does not affect the amount of flux passing through it. However, the orientation of the surface and the direction of the flow vector can affect the magnitude and direction of the flux through the surface.

5. What are some real-world applications of flux through a surface?

Flux through a surface is used in many fields, including physics, engineering, and environmental science. Some examples of its applications include calculating the amount of heat transfer in a building, determining the rate of diffusion in a chemical reaction, and measuring the flow of pollutants through a body of water.

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