Let S be the part of the plane 2x + y + z = 2 that lies inside the cylinder x^2 + y^2 = 1, oriented
upward. Let ⃗ F = (y − z)ˆi + (z − x)ˆj + (x − y)ˆk be a vector field. Evaluate the line integral of ⃗ F around
the boundary of S.

To evaluate the line integral of the vector field \(\mathbf{F} = (y - z)\mathbf{i} + (z - x)\mathbf{j} + (x - y)\mathbf{k}\) around the boundary of the surface \(S\), where \(S\) is the part of the plane \(2x + y + z = 2\) that lies inside the cylinder \(x^2 + y^2 = 1\), we can use **Stokes' Theorem**.

### Step 1: Applying Stokes' Theorem
Stokes' Theorem relates the line integral of a vector field around a closed curve \(C\) to the surface integral of the curl of the vector field over the surface \(S\) bounded by \(C\). Mathematically, it is given by:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
\]
where:
- \(C\) is the boundary of the surface \(S\),
- \(d\mathbf{r}\) is the differential vector along \(C\),
- \(\nabla \times \mathbf{F}\) is the curl of the vector field \(\mathbf{F}\),
- \(d\mathbf{S}\) is the vector area element of the surface \(S\).

### Step 2: Calculate the Curl of \(\mathbf{F}\)
First, we compute the curl of the vector field \(\mathbf{F}\):
\[
\nabla \times \mathbf{F} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
y - z & z - x & x - y
\end{vmatrix}
\]

Expanding this determinant:
\[
\nabla \times \mathbf{F} = \mathbf{i} \left(\frac{\partial (x - y)}{\partial y} - \frac{\partial (z - x)}{\partial z}\right) 
+ \mathbf{j} \left(\frac{\partial (y - z)}{\partial z} - \frac{\partial (x - y)}{\partial x}\right) 
+ \mathbf{k} \left(\frac{\partial (z - x)}{\partial x} - \frac{\partial (y - z)}{\partial y}\right)
\]

Evaluating these partial derivatives:
\[
\nabla \times \mathbf{F} = \mathbf{i} \left(-1 - (-1)\right) + \mathbf{j} \left(-1 - (-1)\right) + \mathbf{k} \left(-1 - (-1)\right)
\]
\[
\nabla \times \mathbf{F} = \mathbf{i}(0) + \mathbf{j}(0) + \mathbf{k}(0) = \mathbf{0}
\]

Thus, the curl of \(\mathbf{F}\) is zero:
\[
\nabla \times \mathbf{F} = \mathbf{0}
\]

### Step 3: Evaluate the Surface Integral
Since \(\nabla \times \mathbf{F} = \mathbf{0}\), the surface integral also evaluates to zero:
\[
\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S \mathbf{0} \cdot d\mathbf{S} = 0
\]

### Conclusion
By Stokes' Theorem, the line integral of \(\mathbf{F}\) around the boundary \(C\) of the surface \(S\) is zero:
\[
\oint_C \mathbf{F} \cdot d\mathbf{r} = 0
\]

Would you like to explore the problem further or do you have any questions? Here are some related questions you might find interesting:

1. What if the vector field \(\mathbf{F}\) was different, say \(\mathbf{F} = (x, y, z)\)? How would that affect the line integral?
2. How do we apply Stokes' Theorem to a surface that's not planar?
3. What is the physical interpretation of the curl of a vector field?
4. How would the result change if the surface \(S\) were oriented downward?
5. Can you derive Stokes' Theorem from first principles?

**Tip:** Remember that when using Stokes' Theorem, verifying that the curl of the vector field is zero can greatly simplify the computation.

Learn how to evaluate the line integral of the vector field F = (y - z)i + (z - x)j + (x - y)k around the boundary of the surface S. Utilizes Stokes' Theorem to relate the line integral to the surface integral of the curl of the vector field over S. Suitable for advanced undergraduate students studying vector calculus and applications of Stokes' Theorem.

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