Use Stokes' Theorem to evaluatef fs curl F dS where F(x,y,z) = %3D and S is the portion of the sphere x² + y² + z² = 5 where 1 < z< V5. Assume S is positively oriented. Show all supporting work. %3D
Use Stokes' Theorem to evaluatef fs curl F dS where F(x,y,z) = %3D and S is the portion of the sphere x² + y² + z² = 5 where 1 < z< V5. Assume S is positively oriented. Show all supporting work. %3D
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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![**Problem Statement**
Use Stokes' Theorem to evaluate the surface integral
\[ \iint_S \text{curl} \, \mathbf{F} \cdot dS \]
where
\[ \mathbf{F}(x, y, z) = \left\langle xy, y, \sec(x^3y^4z) \right\rangle \]
and \( S \) is the portion of the sphere \( x^2 + y^2 + z^2 = 5 \) where \( 1 \leq z \leq \sqrt{5} \). Assume \( S \) is positively oriented. Show all supporting work.
### Explanation
This problem requires the use of Stokes' Theorem to evaluate a surface integral. Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over the boundary curve \( \partial S \). Mathematically, Stokes' Theorem is expressed as:
\[ \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \]
#### Given Data:
- **Vector Field** \( \mathbf{F}(x, y, z) = \left\langle xy, y, \sec(x^3y^4z) \right\rangle \)
- **Surface \( S \)**: Part of the sphere \( x^2 + y^2 + z^2 = 5 \) between \( 1 \leq z \leq \sqrt{5} \)
- **Assumption**: \( S \) is positively oriented.
### Steps to Solve:
1. **Find Curl of Vector Field \( \mathbf{F} \)**
Calculate \( \nabla \times \mathbf{F} \).
2. **Parameterize Surface \( S \)**
Represent the surface \( S \) for calculations.
3. **Find Boundary Curve \( \partial S \)**
Describe the boundary curve of surface \( S \).
4. **Compute Line Integral over \( \partial S \)**
Integrate \( \mathbf{F} \) over the boundary curve \( \partial S \) to solve the problem.
5. **Show All Supporting Work**
Provide all calculations and steps clearly to support the final](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bda9c1d-08ea-450e-a130-838d16131279%2F20789469-6721-4c7b-8399-bfedf260db24%2Fvyz96ah_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Use Stokes' Theorem to evaluate the surface integral
\[ \iint_S \text{curl} \, \mathbf{F} \cdot dS \]
where
\[ \mathbf{F}(x, y, z) = \left\langle xy, y, \sec(x^3y^4z) \right\rangle \]
and \( S \) is the portion of the sphere \( x^2 + y^2 + z^2 = 5 \) where \( 1 \leq z \leq \sqrt{5} \). Assume \( S \) is positively oriented. Show all supporting work.
### Explanation
This problem requires the use of Stokes' Theorem to evaluate a surface integral. Stokes' Theorem relates a surface integral over a surface \( S \) to a line integral over the boundary curve \( \partial S \). Mathematically, Stokes' Theorem is expressed as:
\[ \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \]
#### Given Data:
- **Vector Field** \( \mathbf{F}(x, y, z) = \left\langle xy, y, \sec(x^3y^4z) \right\rangle \)
- **Surface \( S \)**: Part of the sphere \( x^2 + y^2 + z^2 = 5 \) between \( 1 \leq z \leq \sqrt{5} \)
- **Assumption**: \( S \) is positively oriented.
### Steps to Solve:
1. **Find Curl of Vector Field \( \mathbf{F} \)**
Calculate \( \nabla \times \mathbf{F} \).
2. **Parameterize Surface \( S \)**
Represent the surface \( S \) for calculations.
3. **Find Boundary Curve \( \partial S \)**
Describe the boundary curve of surface \( S \).
4. **Compute Line Integral over \( \partial S \)**
Integrate \( \mathbf{F} \) over the boundary curve \( \partial S \) to solve the problem.
5. **Show All Supporting Work**
Provide all calculations and steps clearly to support the final
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