REFER TO IMAGE ---### Assignment: Applying Stokes' Theorem**Task Description:**Use Stokes' Theorem to compute the line integral\[ \oint_C \mathbf{F} \cdot d\mathbf{r} \]where \(C\) is the curve formed by the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(z = 1\). You are given that the curl of \(\mathbf{F}\) is:\[ \text{curl} \, \mathbf{F} = \langle xz, -yz, x^2 + y^2 \rangle. \]**Background:**Stokes' Theorem relates a surface integral of the curl of a vector field over a surface \(\Sigma\) to the line integral of the vector field over its boundary curve \(C\):\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_{\Sigma} (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \]Here, \(\nabla \times \mathbf{F}\) denotes the curl of \(\mathbf{F}\), and \(d\mathbf{S}\) is the vector area element of the surface \(\Sigma\).**Instructions:**1. Recognize the curve \(C\) formed by the intersection of the cylinder \(x^2 + y^2 = 1\) and the plane \(z = 1\).2. Identify the surface \(\Sigma\) with boundary \(C\).3. Compute the surface integral using the given curl of \(\mathbf{F}\).**Key Points:**- The curve \(C\) is a circle of radius 1 in the plane \(z = 1\).- The surface \(\Sigma\) can be taken as the planar disk \(x^2 + y^2 \leq 1\) at \(z = 1\).Below is the detailed solution to guide you through each step.---

Name: REFER TO IMAGE ---### Assignment: Applying Stokes' Theorem**Task Descr
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: REFER TO IMAGE ---### Assignment: Applying Stokes' Theorem**Task Description:**Use Stokes' Theorem to compute the line integral\[ \oint_C \mathbf{F} \cdot d\mathbf{r} \]where \(C\) is the curve formed by the intersection of the cylinder \(x^2 + y^2 =