Question

Use Stokes's Theorem to evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). Use a computer algebra system to verify your results. In each case, \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=4 x z \mathbf{i}+y \mathbf{j}+4 x y \mathbf{k}\) \(S: z=9-x^{2}-y^{2}, \quad z \geq 0\)

Short answer

The solution to the exercise using Stokes' theorem, after calculating and simplifying the surface integral, gives the numerical value of the original line integral. The precise numeric result depends on meticulous calculation of the surface integral.

Step-by-step solution

Compute the Curl of \(\mathbf{F}\)

The curl of the vector field \( \mathbf{F} \) is defined as: \n\n\[ \nabla \times \mathbf{F} = ( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} ) \times (4xz, y, 4xy) \] \n\nAfter calculating the cross product, the curl of \( \mathbf{F} = (0, 4x-4z, 4x) \)

Parameterize the surface \(S\)

To express the surface \(S\) in a simple form, we can use circular cylindrical coordinates (r, \(\phi\), z). Hence, \(S: z=9-r^{2}, \quad z \geq 0\), where \(x = rcos(\phi)\) and \(y = rsin(\phi)\), with \(0 \leq r \leq 3\), and \(0 \leq \phi \leq 2\pi\)

Compute the surface integral using Stokes's Theorem

Using Stokes's Theorem, the line integral can be transformed into a surface integral: \n\n\[ \int_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \] \n\nWhere \(d\mathbf{S}\) is the surface element. Given the parametrization of \(S\), we obtain for \(d\mathbf{S} = (rcos(\phi), rsin(\phi), 9 - r^2)dr d\phi\). After calculating the dot product and integrating over the domain, one gets the value of the original line integral.