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. $$ \begin{aligned} &\mathbf{F}(x, y, z)=x y \mathbf{i}+4 y \mathbf{j}+x z \mathbf{k}\\\ &S: x^{2}+y^{2}+z^{2}=9 \end{aligned} $$

Short answer

The outward flux of the vector field \(\mathbf{F}(x, y, z) = x y \mathbf{i} + 4 y \mathbf{j} + x z \mathbf{k}\) through the surface \(x^{2}+y^{2}+z^{2}=9\) is \(36π\).

Step-by-step solution

Finding Divergence of F

We need to calculate the divergence of the vector field. For a vector field \(\mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\), the divergence is given by \(\nabla \cdot \mathbf{F} = \partial P / \partial x + \partial Q / \partial y + \partial R / \partial z\). Using the given \(\mathbf{F}(x, y, z) = x y \mathbf{i} + 4 y \mathbf{j} + x z \mathbf{k}\), the divergence \(\nabla \cdot \mathbf{F} = \partial (x y) / \partial x + \partial (4 y) / \partial y + \partial (x z) / \partial z = y + 4 + 0 = y + 4\).

Setting up Triple Integral

According to the Divergence Theorem, the outward flux of \(\mathbf{F}\) is given by the triple integral of \(\nabla \cdot \mathbf{F}\) over the volume V enclosed by the surface S. In this case, V is a sphere of radius 3. To set up the triple integral, it's helpful to use spherical coordinates (r, θ, φ). In spherical coordinates, the triple integral becomes \(\iiint_V (y + 4) r^2 \sin(φ) d r d θ dφ\). In spherical coordinates, \(y = r \sin(φ) \cos(θ)\), so we can replace y in the integral. Also, limits of r, θ, and φ are from 0 to 3, 0 to \(2\pi\), and 0 to \(\pi\), respectively.

Evaluating the Triple Integral

Now, we substitute \(y = r \sin(φ) \cos(θ)\) in the triple integral and solve it. \(\iiint_V ((r \sin(φ) \cos(θ)) + 4) r^2 \sin(φ) d r d θ dφ = \\ \iiint_V (r^3 \sin^2(φ) \cos(θ) + 4r^2 \sin(φ)) d r d θ dφ = \\ \int_{0}^3 \int_{0}^{2π} \int_{0}^π (r^3 \sin^2(φ) \cos(θ) + 4r^2 \sin(φ)) dφ dθ dr = \\ 4π \int_{0}^3 r^2 dr \\= 4π [r^3 / 3]_0^3 \\= 4π (3^3 / 3 ) = 36π\).

Verification

It is instructed to use a computer algebra system to verify the results. A computer algebra system such as Mathematica or Maple can precisely perform the divergence and the triple integral calculations. The value obtained from computer computations should agree with the hand calculations.