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
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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