**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.### ExplanationThis 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

Answered: Image /qna-images/answer/20789469-6721-4c7b-8399-bfedf260db24.jpg

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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Question

**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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