27. Gauss' Theorem

Let \(V\) be a nice solid region in \(\mathbb{R}^3\) with a nice properly oriented boundary, \(\partial V\), and let \(\vec F\) be a nice vector field on \(V\). Then \[ \iiint_V \vec\nabla\cdot\vec F\,dV =\iint_{\partial V} \vec F\cdot d\vec S \] The outer boundary must be oriented outward while any inner boundaries must be oriented inward. This means that all pieces of the boundary are oriented away from the solid.

Homework

1. Verify Gauss' Theorem \[ \iiint_P \vec\nabla\cdot\vec F\,dV =\iint_{\partial P} \vec F\cdot d\vec S \] for the vector field \(\vec{F}=\big\langle xz,yz,z^2\big\rangle\) and the volume above the paraboloid, \(P\): \(z=x^2+y^2\) for \(z\leq 4\) and below the disk, \(D\): \(x^2+y^2\leq 4\) with \(z=4\).

   

   1. Compute the divergence in rectangular coordinates. Then choose a coordinate system for the integral, and express the divergence and \(dV\) in those coordinates. Then compute the volume integral.

   2. Parameterize each piece of the surface. Compute tangent and normal vectors. Check the orientation. Evaluate \(\vec{F}\) on each surface and compute the flux through each surface. Finally add the surface integrals and compare the answer with part (a).

2. Verify Gauss' Theorem \[ \iiint_C \vec\nabla\cdot\vec F\,dV =\iint_{\partial C} \vec F\cdot d\vec S \] for the vector field \(\vec{F}=\langle xz^2,yz^2,z(x^2+y^2)\rangle\) and the volume above the cylinder, \(C\): \(x^2+y^2 \le 4\) for \(1 \le z \le 4\).

   

   1. Compute the divergence in rectangular coordinates. Then choose a coordinate system for the integral, and express the divergence and \(dV\) in those coordinates. Then compute the volume integral.

   2. Parameterize each piece of the surface. Compute tangent and normal vectors. Check each orientation. Evaluate \(\vec{F}\) on each surface and compute the flux through each surface. Finally add the surface integrals and compare the answer with part (a).