Calculating Flux of F across Surface S | Flux Homework Help

[kthejohn]

Homework Statement

Let S be the surface z=x+y, 0≤x≤1, 0≤y≤1. Find the upward flux of the vector field F =<z,x,y > across S.

Homework Equations

∫∫_SFndS

The Attempt at a Solution

parametrize:
x=x
y=y
z=x+y
r_x=<1,0,1> r_y=<0,1,1>

 

[HallsofIvy]

Homework Statement

Let S be the surface z=x+y, 0≤x≤1, 0≤y≤1. Find the upward flux of the vector field F =<z,x,y > across S.

Homework Equations

∫∫_SFndS

The Attempt at a Solution

parametrize:
x=x
y=y
z=x+y
r_x=<1,0,1> r_y=<0,1,1>

Excellent! Now take the cross product:
$$\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right |= -\vec{i}- \vec{j}+ \vec{k}$$
where the order of multiplication has been chosen to give positive $\vec{k}$ for upward flow. Now
$$d\vec{S}= \vec{n} dS= (-\vec{i}- \vec{j}+ \vec{k})dxdy$$
is the vector differential of surface area. Integrate the dot product of that with the flux vector,
$$z\vec{i}+ x\vec{j}+ y\vec{k}= (x+ y)\vec{i}+ x\vec{j}+ y\vec{k}$$.