Find the field flow using integrals

[kliker]

Homework Statement

find the field's F flow where F = (y,-z,2) through the surface S where S is 4x+2y+z=6 and x,y,z>=0

The Attempt at a Solution

we know that

[PLAIN]http://img195.imageshack.us/img195/4971/asdqg.gif

we know F

N is the derivative of 4x+2y+z-6 = g(x)

actually it's N = (dg/dx,dg/dy,dg/dz)

now what i want to ask is how to find the limits of the integral i mean from what point to what point i should integrate

what I can think of is because x,y,z are bigger than 0 hence they are possitive

we will have the same points for dy and dx which will be from 0 to +oo

so can i just say

[PLAIN]http://img514.imageshack.us/img514/351/asdd.gif

find the integral and then take the limit using xn -> oo and yn -> oo?

thanks in advance

edit: but i have a z, what am i going to do with z?

 

[radou]

You need to evaluate a surface integral. Rewrite the equation of the plane in the form z = f(x, y), plug it into your integral, and find the triangular region over which you need to integrate, together with the appropriate boundaries of integration.

 

[radou]

By the way, applying the divergence theorem yields the answer very quickly.

 

[HallsofIvy]

4x+ 2y+ z= 6 is a plane. Since "x, y, z>= 0", that is, all coordinates must be non-negative, you want the portion of the plane that is in the first octant. It's boundary consists of points where some of x, y, z are equal to 0. Specifically in the xy-plane, where z= 0, the triangular region radou referred to has boundaries x= 0, y= 0, and 4x+ 2y= 6 so that y= 2x- 3. y will also be 0 when 2x- 3=0 or x= 3/2.

x goes from 0 to 3/2 and, for every x, y goes from 0 to 2x- 3.

 

[kliker]

thanks for your help guys

one last question

should I replace z = 6-2y-4x if i have a z in the integral?

 

[LCKurtz]

By the way, applying the divergence theorem yields the answer very quickly.

No, it doesn't. You don't have an enclosed volume.

thanks for your help guys

one last question

should I replace z = 6-2y-4x if i have a z in the integral?

Yes. But I have a couple of questions for you. First, did you check the orientation of the surface and do you know you are calculating the flow in the proper direction?

Second, you apparently got your normal vector by taking the gradient of the plane equation, which gives the coefficients. But [flux] surface integrals are of the form

$$\int\int_R \vec F \cdot \hat n\ dS$$

using a unit normal, which your N isn't. Now it turns out that your integrand is correct but my question to you is do you know why or did you just get lucky? Why did you not use the unit normal? Are you using something you didn't tell us?

 

[HallsofIvy]

By the way, applying the divergence theorem yields the answer very quickly.

No, it doesn't. You don't have an enclosed volume.

But Stoke's theorem could convert it into an integration around the bounding triangle. That probably would be harder, not simpler!

 

[kliker]

No, it doesn't. You don't have an enclosed volume.
Yes. But I have a couple of questions for you. First, did you check the orientation of the surface and do you know you are calculating the flow in the proper direction?

Second, you apparently got your normal vector by taking the gradient of the plane equation, which gives the coefficients. But [flux] surface integrals are of the form

$$\int\int_R \vec F \cdot \hat n\ dS$$

using a unit normal, which your N isn't. Now it turns out that your integrand is correct but my question to you is do you know why or did you just get lucky? Why did you not use the unit normal? Are you using something you didn't tell us?

We ve learned in Math that field flow has this kind of integral, that's why I used it

the reason i used simple F and N is because i don't know how to create a vector F and N on my keyboard

 

[LCKurtz]

Yes. But I have a couple of questions for you. First, did you check the orientation of the surface and do you know you are calculating the flow in the proper direction?

Second, you apparently got your normal vector by taking the gradient of the plane equation, which gives the coefficients. But [flux] surface integrals are of the form

$$\int\int_R \vec F \cdot \hat n\ dS$$

using a unit normal, which your N isn't. Now it turns out that your integrand is correct but my question to you is do you know why or did you just get lucky? Why did you not use the unit normal? Are you using something you didn't tell us?

We ve learned in Math that field flow has this kind of integral, that's why I used it

the reason i used simple F and N is because i don't know how to create a vector F and N on my keyboard

I wasn't asking because of the typing. I was asking because you didn't use a unit vector in your calculation. Your N is not a unit vector. And dS is not dxdy. I was wondering if you could explain about that. And you didn't answer the question about the orientation of the surface. Which way is it oriented and does your choice of normal agree with it?