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. \(\mathbf{F}(x, y, z)=z^{2} \mathbf{i}+y \mathbf{j}+x z \mathbf{k}\) \(S: z=\sqrt{4-x^{2}-y^{2}}\)

Short answer

The value of the given line integral, evaluated using Stoke's Theorem, is 0.

Step-by-step solution

Compute the Curl of the Vector Field

The curl of the vector field is given by \(\nabla \times \mathbf{F}\). Here \(\nabla\) is the vector operator \((\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})\) and cross product with \(\mathbf{F} = (z^{2}, y, xz)\) gives us Curl \(\mathbf{F} = (1, -2z, -1)\).

Set Up the Surface Integral

By Stoke's Theorem, we can express the given line integral in terms of a surface integral \(\int_{S}(\nabla \times \mathbf{F}) \cdot d\mathbf{S}\). Where \(\mathbf{S} = \sqrt{4-x^{2}-y^{2}}\). Evaluation of surface integral follows the formula \(\int_{S} \mathbf{F} \cdot d\mathbf{S} = \iint_{D} \mathbf{F} \cdot ( \mathbf{N} \, dS)\), where N is the outward unit normal vector to the surface and D is the projection of S in the xy-plane.

Compute the Normal Vector

The outward pointing normal vector for the surface S is given by \(\mathbf{N} = \frac{(-F_x, -F_y, 1)}{\sqrt{1 + F_x^2 + F_y^2}} = \frac{(-(-x/\sqrt{4 - x^2 - y^2}),-(-y/\sqrt{4 - x^2 - y^2}),1)}{\sqrt{1 + x^2/(4 - x^2 - y^2) + y^2/(4 - x^2 - y^2)}} = (x/2, y/2, 1/2)\)

Evaluate the Surface Integral

Now we're able to put it all together and compute the surface integral. First, the dot product of the Curl \(\mathbf{F}\) and surface Norm N of the surface gives \((1, -2z, -1) \cdot (x/2, y/2, 1/2) = x/2 -z -1/2 \), with ds = rdrd\[theta\] and r ranges from 0 to 2 and \[theta\] from 0 to 2\pi. Hence, \(\iint_{D} \mathbf{F} \cdot ( \mathbf{N} \, dS) = \int_{0}^{2\pi} \int_{0}^{2} (x/2-z-1/2) r \, dr d\[\theta\] = \int_{0}^{2\pi} \int_{0}^{2} (2cos\[\theta\]/2 - \sqrt{4-r^2} -1/2) \, dr d\[\theta\] = 0\)