4) Use Stokes theorem to evaluate SF.dr, where с F(x, y, z) = <.xy, yz, *x), where C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1).
4) Use Stokes theorem to evaluate SF.dr, where с F(x, y, z) = <.xy, yz, *x), where C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1).
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
This is Calculus 3
![### Problem Statement:
**4.** Use Stokes' Theorem to evaluate the line integral:
\[ \oint_C \mathbf{F} \cdot d\mathbf{r}, \]
where \(\mathbf{F}(x,y,z) = \langle xy, yz, xz \rangle\), and \(C\) is the triangle with vertices \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).
### Explanation:
Stokes' Theorem is a powerful tool in vector calculus that relates a surface integral over a surface \(S\) to a line integral over its boundary curve \(C\). The theorem is stated mathematically as:
\[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}, \]
where
- \(\nabla \times \mathbf{F}\) is the curl of the vector field \(\mathbf{F}\),
- \(d\mathbf{S}\) is the vector area element of the surface \(S\).
Given:
- The vector field \(\mathbf{F}(x,y,z) = \langle xy, yz, xz \rangle\),
- The vertices of the triangle: \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).
### Steps to Solve:
1. **Parameterize the Surface:**
Determine the surface \(S\) whose boundary is the triangle with the given vertices.
2. **Curl of \(\mathbf{F}\):**
Compute \(\nabla \times \mathbf{F}\).
3. **Surface Normal Vector:**
Determine the normal vector \(d\mathbf{S}\) for the triangle.
4. **Evaluate the Surface Integral:**
Integrate \((\nabla \times \mathbf{F}) \cdot d\mathbf{S}\) over the surface \(S\).
### Additional Information:
- **Graphs/Diagrams:**
This problem may include diagrams of the triangular surface, vector field \(\mathbf{F}\), and the direction of the normal vector to aid in visualization.
- **Figures and Visual Aids:**
A sketch showing the triangle](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a1af84-f79e-495e-9bf6-e155c46aa5b8%2F9bda66e7-ae97-4295-8cd9-0cf83b0bed41%2Fq4fit4oa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
**4.** Use Stokes' Theorem to evaluate the line integral:
\[ \oint_C \mathbf{F} \cdot d\mathbf{r}, \]
where \(\mathbf{F}(x,y,z) = \langle xy, yz, xz \rangle\), and \(C\) is the triangle with vertices \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).
### Explanation:
Stokes' Theorem is a powerful tool in vector calculus that relates a surface integral over a surface \(S\) to a line integral over its boundary curve \(C\). The theorem is stated mathematically as:
\[ \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}, \]
where
- \(\nabla \times \mathbf{F}\) is the curl of the vector field \(\mathbf{F}\),
- \(d\mathbf{S}\) is the vector area element of the surface \(S\).
Given:
- The vector field \(\mathbf{F}(x,y,z) = \langle xy, yz, xz \rangle\),
- The vertices of the triangle: \((1,0,0)\), \((0,1,0)\), and \((0,0,1)\).
### Steps to Solve:
1. **Parameterize the Surface:**
Determine the surface \(S\) whose boundary is the triangle with the given vertices.
2. **Curl of \(\mathbf{F}\):**
Compute \(\nabla \times \mathbf{F}\).
3. **Surface Normal Vector:**
Determine the normal vector \(d\mathbf{S}\) for the triangle.
4. **Evaluate the Surface Integral:**
Integrate \((\nabla \times \mathbf{F}) \cdot d\mathbf{S}\) over the surface \(S\).
### Additional Information:
- **Graphs/Diagrams:**
This problem may include diagrams of the triangular surface, vector field \(\mathbf{F}\), and the direction of the normal vector to aid in visualization.
- **Figures and Visual Aids:**
A sketch showing the triangle
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