Divergence Theorem — Question 6

Question 6

Let $E$ be the first-octant tetrahedron $x+y+z\le 1$, and let $S=\partial E$ be outward oriented. For $$\mathbf F(x,y,z)=\langle x^2,y^2,z^2\rangle,$$ compute the total outward flux.

Tasks

1. Find the divergence.

2. Evaluate the needed coordinate moments over the tetrahedron.

3. Check the answer directly on the slanted face.

Question 6 – Solution

Strategy. Symmetry makes the three first moments equal; the coordinate-plane faces also provide a short direct check.

Step 1: Divergence $$\nabla\cdot\mathbf F=2x+2y+2z.$$ The tetrahedron has volume $1/6$ and centroid $(1/4,1/4,1/4)$, so $$\iiint_E x\,dV=\iiint_E y\,dV=\iiint_E z\,dV =\frac 14\cdot\frac 16=\frac 1{24}.$$

Step 2: Apply the theorem $$\boxed{\iint_S\mathbf F\cdot\mathbf n\,dS =2\left(\frac 1{24}+\frac 1{24}+\frac 1{24}\right)=\frac 14}.$$

Step 3: Direct check On each coordinate face the corresponding field component is zero, so those fluxes vanish. On $z=1-x-y$, the outward vector element is $\langle 1,1,1\rangle dA$, giving $$\iint_D\bigl(x^2+y^2+(1-x-y)^2\bigr)\,dA=\frac 14.$$

Verification The direct slanted-face integral agrees with the divergence result and is positive, as expected in the first octant.