Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (-1, 1, 1), (2, 1, −5), (2, 3, -11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1, -5) from (2, 1, 5) to (2, 3, -11) from (2, 3, 11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁(t) = S₂ (t) S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) tE [3, 4) UE [-1, 2], VE [1, 3]
Let S be the portion of the plane 2x + 3y + z = 2 lying between the points (-1, 1, 1), (2, 1, −5), (2, 3, -11), and (-1, 3, -5). Find parameterizations for both the surface S and its boundary S. Be sure that their respective orientations are compatible with Stokes' theorem. from (-1, 1, 1) to (2, 1, -5) from (2, 1, 5) to (2, 3, -11) from (2, 3, 11) to (-1, 3, -5) from (-1, 3, 5) to (-1, 1, 1) boundary S₁(t) = S₂ (t) S3(t) = S4(t) = Φ(u, v) = te [0, 1) te [1, 2) te [2, 3) tE [3, 4) UE [-1, 2], VE [1, 3]
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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![### Parameterization of Planes and Boundaries
**Problem Statement:**
Let \( S \) be the portion of the plane \( 2x + 3y + z = 2 \) lying between the points \((-1, 1, 1)\), \((2, 1, -5)\), \((2, 3, -11)\), and \((-1, 3, -5)\). Find parameterizations for both the surface \( S \) and its boundary \( \partial S \). Ensure that their respective orientations are compatible with Stokes' theorem.
### Parameterization of Boundary Segments
1. **From \((-1, 1, 1)\) to \((2, 1, -5)\)**
\[ S_1(t) = \boxed{} \]
where \( t \in [0, 1] \).
2. **From \((2, 1, -5)\) to \((2, 3, -11)\)**
\[ S_2(t) = \boxed{} \]
where \( t \in [1, 2] \).
3. **From \((2, 3, -11)\) to \((-1, 3, -5)\)**
\[ S_3(t) = \boxed{} \]
where \( t \in [2, 3] \).
4. **From \((-1, 3, -5)\) to \((-1, 1, 1)\)**
\[ S_4(t) = \boxed{} \]
where \( t \in [3, 4] \).
### Parameterization of Surface
**Boundary**
\[ \Phi(u, v) = \boxed{} \]
where \( u \in [-1, 2] \) and \( v \in [1, 3] \).
### Explanation of Graphs/Diagrams
**Graph/Diagram Description:**
- **Segment Graphs:** Each boundary segment \( S_1(t), S_2(t), S_3(t), \) and \( S_4(t) \) represents a linear path between the specified points. These segments are parameterized by \( t \), defining their locations in 3D space over a specific interval.
- **Surface Parameterization:** The surface parameter \( \Phi(u, v) \) describes a mapping from a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f1c68a1-c113-41cc-b0cf-42f2666e5687%2F37d50002-b7ce-4927-bdf0-5689a7fbc783%2F8dbj93_processed.png&w=3840&q=75)
Transcribed Image Text:### Parameterization of Planes and Boundaries
**Problem Statement:**
Let \( S \) be the portion of the plane \( 2x + 3y + z = 2 \) lying between the points \((-1, 1, 1)\), \((2, 1, -5)\), \((2, 3, -11)\), and \((-1, 3, -5)\). Find parameterizations for both the surface \( S \) and its boundary \( \partial S \). Ensure that their respective orientations are compatible with Stokes' theorem.
### Parameterization of Boundary Segments
1. **From \((-1, 1, 1)\) to \((2, 1, -5)\)**
\[ S_1(t) = \boxed{} \]
where \( t \in [0, 1] \).
2. **From \((2, 1, -5)\) to \((2, 3, -11)\)**
\[ S_2(t) = \boxed{} \]
where \( t \in [1, 2] \).
3. **From \((2, 3, -11)\) to \((-1, 3, -5)\)**
\[ S_3(t) = \boxed{} \]
where \( t \in [2, 3] \).
4. **From \((-1, 3, -5)\) to \((-1, 1, 1)\)**
\[ S_4(t) = \boxed{} \]
where \( t \in [3, 4] \).
### Parameterization of Surface
**Boundary**
\[ \Phi(u, v) = \boxed{} \]
where \( u \in [-1, 2] \) and \( v \in [1, 3] \).
### Explanation of Graphs/Diagrams
**Graph/Diagram Description:**
- **Segment Graphs:** Each boundary segment \( S_1(t), S_2(t), S_3(t), \) and \( S_4(t) \) represents a linear path between the specified points. These segments are parameterized by \( t \), defining their locations in 3D space over a specific interval.
- **Surface Parameterization:** The surface parameter \( \Phi(u, v) \) describes a mapping from a
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