Question

In Exercises \(5-16,\) use the Divergence Theorem to evaluate \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S\) and find the outward flux of \(F\) through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. $$ \begin{aligned} &\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}-y z \mathbf{k}\\\ &S: z=\sqrt{a^{2}-x^{2}-y^{2}}, z=0 \end{aligned} $$

Short answer

The flux of F through the surface S is obtained by calculating the above described integral. For a concrete numerical answer, We need to carry out the calculations which is best done with a computer algebra system.

Step-by-step solution

Get the expression for the volume

The volume V under the surface S is bounded by \(z=0\) and \(z=\sqrt{a^{2}-x^{2}-y^{2}}\). Thus, we conclude that the volume V is a hemisphere of radius a.

Calculate the Divergence of F

The divergence of F is given by \(\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (x y) + \frac{\partial}{\partial y} (y z) - \frac{\partial}{\partial z} (y z) = y + z - z = y.\)

Transform the Surface Integral into a Volume Integral

According to the Divergence Theorem, the flux of \(F\) through the surface S is equal to the volume integral of the divergence of \(F\) over the volume V. Thus, \(\int_{S} \int \mathbf{F} \cdot \mathbf{N} d S = \int_{V} \nabla \cdot \mathbf{F} d V = \int_{V} y d V.\)

Evaluate the Volume Integral

The volume integral can be evaluated in spherical coordinates. We take the volume element in spherical coordinates as \(d V = r^{2} \sin \theta dr d \theta d \phi.\) Then, because the volume V is a hemisphere, \(r\) ranges from \(0\) to \(a\), \(\theta\) ranges from \(0\) to \(\pi/2\), and \(\phi\) ranges from \(0\) to \(2 \pi\). And in spherical coordinates, \(y = r \sin \theta \cos \phi.\) Substituting these, the volume integral becomes \(\int_{0}^{2 \pi} \int_{0}^{\pi/2} \int_{0}^{a} r^{3} \sin^{2} \theta \cos \phi dr d \theta d \phi.\) Evaluate the integral.

Result Verification

One can now use a computer algebra system to verify the results.