Surface Integrals of Vector Fields — Question 6

Question 6

Let $E$ be the tetrahedron in the first octant bounded by the coordinate planes and $x+y+z=1$. Let $S=\partial E$ carry the outward orientation, and let $$\mathbf F(x,y,z)=\langle x,y,z\rangle.$$ Compute the total outward flux directly by summing the four faces.

Tasks

1. Determine the flux through the three coordinate-plane faces.

2. Parametrize the slanted face with its outward orientation.

3. Sum the face contributions and verify the sign.

Question 6 – Solution

Strategy. The field has zero normal component on each coordinate face, leaving only the slanted triangular face.

Step 1: Coordinate faces On $x=0$, the outward normal is $-\mathbf i$ and $\mathbf F\cdot(-\mathbf i)=-x=0$. Similarly, the flux densities on $y=0$ and $z=0$ vanish. These three fluxes are all zero.

Step 2: Slanted face Write $z=1-x-y$ over $$D=\{(x,y):x\ge 0,\ y\ge 0,\ x+y\le 1\}.$$ The outward side points away from the origin, and its vector element is $$\boxed{\mathbf n\,dS=\langle 1,1,1\rangle\,dx\,dy}.$$

On this face, $$\mathbf F\cdot\langle 1,1,1\rangle=x+y+z=1.$$

Step 3: Integrate and sum $$\iint_{\partial E}\mathbf F\cdot\mathbf n\,dS =0+\iint_D1\,dA =\boxed{\frac 12}.$$

Verification The field points away from the origin and is tangent to the coordinate faces, so all nonzero flux must leave through the slanted face and must be positive.