REFER TO IMAGE # 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 Approach1. 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 Theorem1. **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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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.

Trigonometry (MindTap Course List)

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ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.6: Parametric Equations

Problem 5ECP: Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle...

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REFER TO IMAGE

# 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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