# Flow through a surface

1. Aug 23, 2011

### Lindsayyyy

1. The problem statement, all variables and given/known data

given is the following field $$F(x,y,z)=(y,xz,0)$$ (F is a vector field) and the borders of the surface are: $$0<x<1 ;y=x ; 0<z<1$$ the < should be less equal but I don't know how to do the sign in latex, sorry. The normal vector is given as $$n=(a,b,c); b<0$$

I shall calculate the flow through the surface
2. Relevant equations

the formula for the flow, can't type in in latex, sorry, but I think you know which one I mean (flow= integral F*n*dS)

3. The attempt at a solution
I need to find my dS, problem here I have, I don't know how to do this exactly. I tried to parametrize it, but I'm not sure how to do it, because I have a function in my borders. So I think y max equals 1 because of the requirements for x. Can anyone help me?

2. Aug 23, 2011

### Quinzio

If you try to use $y=x$ anywhere you can, it will be all quite simple.
Which parametrization did you try ?

3. Aug 23, 2011

### Lindsayyyy

I tried x,x,z not sure though.

4. Aug 23, 2011

### HallsofIvy

Staff Emeritus
That surface itself is defined by y= x. The position vector for any point on that surface is r(x, z)= <x, x, z>. Two tangent vectors, at any point are rx= < 1, 1, 0> and another is rz= <0, 0, 1>. The vector differential of surface area is given by $d\vec{S}= \vec{v_s}\times\vec{v_t} dsdt$. Write your vector function in terms of s and t and integrate the dot product $\vec{F}\cdot d\vec{S}$

Last edited: Aug 23, 2011
5. Aug 23, 2011

### Lindsayyyy

Thank you very much, that's what I did after I got the parametrisation. I wasn't sure about it though.