5. Let S be the part of the paraboloid z = 9 – a² – y² (with upward orientation) that lies above the plane z = 0. Let F = (r, 2.xz, xy). (a) Set up an iterated integral to find /| curl F - dS directly.
5. Let S be the part of the paraboloid z = 9 – a² – y² (with upward orientation) that lies above the plane z = 0. Let F = (r, 2.xz, xy). (a) Set up an iterated integral to find /| curl F - dS directly.
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![# Surface Integrals and the Curl of a Vector Field
## Problem Statement
### Given:
Let \( S \) be the part of the paraboloid \( z = 9 - x^2 - y^2 \) (with upward orientation) that lies above the plane \( z = 0 \). Let \( \mathbf{F} = \langle x, 2xz, xy \rangle \).
### Tasks:
(a) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) directly.
(b) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) using an appropriate theorem.
## Solution Steps
### Part (a) Direct Approach
1. To find the iterated integral directly, identify the surface \( S \).
2. **Surface \( S \)**:
- The surface is part of the paraboloid \( z = 9 - x^2 - y^2 \).
- \( S \) has an upward normal.
3. **Vector Field \( \mathbf{F} \)**:
- \( \mathbf{F} = \langle x, 2xz, xy \rangle \).
4. **Curl of \( \mathbf{F} \)**:
- Evaluate \( \nabla \times \mathbf{F} \).
\[
\nabla \times \mathbf{F} = \nabla \times \langle x, 2xz, xy \rangle
\]
5. **Surface Element \( d\mathbf{S} \)** for Paraboloid:
- Calculate \( d\mathbf{S} \) for \( z = 9 - x^2 - y^2 \).
6. **Integral Form**:
- Set up the iterated integral involving \( (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \).
### Part (b) Using an Appropriate Theorem
1. **Use Stokes' Theorem**:
- Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral around the boundary curve of the surface.
\[
\iint_S (\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191441f0-66db-457d-b504-dd20fe1485fa%2F97d23e88-8f32-4388-9d2b-9ee5f7b4679a%2Fu1bn31_processed.png&w=3840&q=75)
Transcribed Image Text:# Surface Integrals and the Curl of a Vector Field
## Problem Statement
### Given:
Let \( S \) be the part of the paraboloid \( z = 9 - x^2 - y^2 \) (with upward orientation) that lies above the plane \( z = 0 \). Let \( \mathbf{F} = \langle x, 2xz, xy \rangle \).
### Tasks:
(a) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) directly.
(b) **Set up** an iterated integral to find \( \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \) using an appropriate theorem.
## Solution Steps
### Part (a) Direct Approach
1. To find the iterated integral directly, identify the surface \( S \).
2. **Surface \( S \)**:
- The surface is part of the paraboloid \( z = 9 - x^2 - y^2 \).
- \( S \) has an upward normal.
3. **Vector Field \( \mathbf{F} \)**:
- \( \mathbf{F} = \langle x, 2xz, xy \rangle \).
4. **Curl of \( \mathbf{F} \)**:
- Evaluate \( \nabla \times \mathbf{F} \).
\[
\nabla \times \mathbf{F} = \nabla \times \langle x, 2xz, xy \rangle
\]
5. **Surface Element \( d\mathbf{S} \)** for Paraboloid:
- Calculate \( d\mathbf{S} \) for \( z = 9 - x^2 - y^2 \).
6. **Integral Form**:
- Set up the iterated integral involving \( (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \).
### Part (b) Using an Appropriate Theorem
1. **Use Stokes' Theorem**:
- Stokes' Theorem relates a surface integral of the curl of a vector field to a line integral around the boundary curve of the surface.
\[
\iint_S (\
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