Use the Divergence Theorem to evaluate Verify your answer by evaluating the integral as a triple integral. S: cube bounded by the planes

Question

Use the Divergence Theorem to evaluate  Verify your answer by evaluating the integral as a triple integral.  S: cube bounded by the planes

Quizplus · Accepted Answer

By the Divergence Theorem:
$$\iint_S\vec{F}\cdot\vec{n}\,dS=\iiint_E 	ext{div}\,\vec{F}\,dV$$
where $\vec{n}$ is the outward pointing normal vector to the surface $S$, and $	ext{div}\,\vec{F}$ is the divergence of the vector field $\vec{F}$.
In this case, we have:
$$	ext{div}\,\vec{F}=2x+2y+2z$$
which means that:
$$\iiint_E 	ext{div}\,\vec{F}\,dV=\int_{-1}^1\int_{-1}^1\int_{-1}^1 (2x+2y+2z)\,dx\,dy\,dz=0$$
since the integrand is an odd function of each variable and the limits of integration are symmetric about zero.
Therefore, the surface integral simplifies to:
$$\iint_S\vec{F}\cdot\vec{n}\,dS=0$$
since the divergence is zero everywhere inside $E$.

Use the Divergence Theorem to Evaluate Verify Your Answer