Calculating Flow through a Surface with a Given Vector Field and Normal Vector

[Lindsayyyy]

Homework Statement

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

Homework 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)

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?

 

[Quinzio]

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

 

[Lindsayyyy]

I tried x,x,z not sure though.

 

[HallsofIvy]

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 r_x= < 1, 1, 0> and another is r_z= <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 $\vec{F}\cdot d\vec{S}$

 

[Lindsayyyy]

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