Question

Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $\mathbf{F}(x, y, z)=2 x \mathbf{i}-y z \mathbf{j}+z^{2} \mathbf{k} ;$ the surface $\sigma$ is the paraboloid $z=x^{2}+y^{2}$ capped by the disk $x^{2}+y^{2} \leq 1$ in the plane $z=1.$

          Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral.
$\mathbf{F}(x, y, z)=2 x \mathbf{i}-y z \mathbf{j}+z^{2} \mathbf{k} ;$ the surface $\sigma$ is the paraboloid $z=x^{2}+y^{2}$ capped by the disk $x^{2}+y^{2} \leq 1$ in the plane $z=1.$

Added by Lisa M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

08/23/2023

Step 1

Let $x = r\cos\theta$, $y = r\sin\theta$, and $z = z$. Then, $dV = r\, dr\, d\theta\, dz$. The limits of integration are $0 \leq r \leq 1$, $0 \leq \theta \leq 2\pi$, and $r^2 \leq z \leq 1$. So, the triple integral becomes: $$\iiint_V (2 + 2z) \, dV = Show more…

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Verify Formula (1) in the Divergence Theorem by evaluating the surface integral and the triple integral. $\mathbf{F}(x, y, z)=2 x \mathbf{i}-y z \mathbf{j}+z^{2} \mathbf{k} ;$ the surface $\sigma$ is the paraboloid $z=x^{2}+y^{2}$ capped by the disk $x^{2}+y^{2} \leq 1$ in the plane $z=1.$