6) Use Stokes' Theorem and Theorem 10 in section 16.7 to find § F · dř. F(x, y, z) = (2y, xz, x + y ). C is oriented counterclockwise as viewed from above and is the intersection of the plane z = y + 2 and the cylinder x² + y² = 1.
6) Use Stokes' Theorem and Theorem 10 in section 16.7 to find § F · dř. F(x, y, z) = (2y, xz, x + y ). C is oriented counterclockwise as viewed from above and is the intersection of the plane z = y + 2 and the cylinder x² + y² = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![#### Problem Statement
**6)** Use Stokes' Theorem and Theorem 10 in section 16.7 to find ∫<sub>C</sub> **F** · d**r**.
\[ \mathbf{F}(x,y,z) = \langle 2y, xz, x+y \rangle \]
**C** is oriented counterclockwise as viewed from above and is the intersection of the plane \( z = y + 2 \) and the cylinder \( x^2 + y^2 = 1 \).
### Explanation:
1. **Stokes' Theorem:**
Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over its boundary curve \( C \):
\[
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
\]
2. **Parameterization of \( C\):**
The curve \( C \) is the intersection of the plane \( z = y + 2 \) and the cylinder \( x^2 + y^2 = 1 \).
Ensure to check section 16.7 for Theorem 10 to see how it complements Stokes' Theorem in the computation of this problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb10c217e-0837-4d3f-89c5-6d9b36b54f81%2F595ea26d-3056-43de-949e-5c29e62d11f6%2Fzgkm564x_processed.png&w=3840&q=75)
Transcribed Image Text:#### Problem Statement
**6)** Use Stokes' Theorem and Theorem 10 in section 16.7 to find ∫<sub>C</sub> **F** · d**r**.
\[ \mathbf{F}(x,y,z) = \langle 2y, xz, x+y \rangle \]
**C** is oriented counterclockwise as viewed from above and is the intersection of the plane \( z = y + 2 \) and the cylinder \( x^2 + y^2 = 1 \).
### Explanation:
1. **Stokes' Theorem:**
Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over its boundary curve \( C \):
\[
\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}
\]
2. **Parameterization of \( C\):**
The curve \( C \) is the intersection of the plane \( z = y + 2 \) and the cylinder \( x^2 + y^2 = 1 \).
Ensure to check section 16.7 for Theorem 10 to see how it complements Stokes' Theorem in the computation of this problem.
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