Evaluate the line integral in Stokes' Theorem to evaluate the surface integral SS(VXF). VXF)•n dS. S Assume that n points in an upward direction. F = (x + y, y+z, z+x); S is the tilted disk enclosed by r(t) = (4 cos t, 6 sint, √20 cost).
Evaluate the line integral in Stokes' Theorem to evaluate the surface integral SS(VXF). VXF)•n dS. S Assume that n points in an upward direction. F = (x + y, y+z, z+x); S is the tilted disk enclosed by r(t) = (4 cos t, 6 sint, √20 cost).
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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**Evaluating the Line Integral in Stokes' Theorem**
To understand how to evaluate the line integral in Stokes' Theorem effectively, we need to evaluate the surface integral provided.
\[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS \]
In this context, we need to make some assumptions and follow certain steps.
1. **Assumptions:**
- Assume that the normal vector \( \mathbf{n} \) points in an upward direction.
2. **Function Definition:**
- The vector field \( \mathbf{F} \) is defined as:
\[ \mathbf{F} = (x + y, y + z, z + x) \]
- The surface \( S \) is a tilted disk. This disk is enclosed by the parametric curve \( \mathbf{r}(t) \), given by:
\[ \mathbf{r}(t) = \left( 4 \cos t, 6 \sin t, \sqrt{20} \cos t \right) \]
In order to compute the required surface integral:
1. **Calculate the Curl of \( \mathbf{F} \):**
- Use the standard curl operation \( \nabla \times \mathbf{F} \) applied to the given vector field \( \mathbf{F} \).
2. **Parametrize the Surface \( S \):**
- Express the surface \( S \) parametrically to accurately relate \( \mathbf{r}(t) \) to the surface.
3. **Determine the Normal Vector \( \mathbf{n} \):**
- The normal vector \( \mathbf{n} \) is assumed to point upwards and should be derived based on the parametrization of the surface.
4. **Compute the Surface Integral:**
- Evaluate the surface integral using the curl of \( \mathbf{F} \) and the unit normal vector \( \mathbf{n} \).
This process should align with the steps outlined in Stokes' Theorem, leading to a thorough understanding and accurate evaluation of the given surface integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7f7e91-311f-40b2-8ef2-058480eb16a0%2Fdd4bbf31-8191-4c82-b50b-f645efc91d8c%2Fq2tyej8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Evaluating the Line Integral in Stokes' Theorem**
To understand how to evaluate the line integral in Stokes' Theorem effectively, we need to evaluate the surface integral provided.
\[ \iint_S (\nabla \times \mathbf{F}) \cdot \mathbf{n} \, dS \]
In this context, we need to make some assumptions and follow certain steps.
1. **Assumptions:**
- Assume that the normal vector \( \mathbf{n} \) points in an upward direction.
2. **Function Definition:**
- The vector field \( \mathbf{F} \) is defined as:
\[ \mathbf{F} = (x + y, y + z, z + x) \]
- The surface \( S \) is a tilted disk. This disk is enclosed by the parametric curve \( \mathbf{r}(t) \), given by:
\[ \mathbf{r}(t) = \left( 4 \cos t, 6 \sin t, \sqrt{20} \cos t \right) \]
In order to compute the required surface integral:
1. **Calculate the Curl of \( \mathbf{F} \):**
- Use the standard curl operation \( \nabla \times \mathbf{F} \) applied to the given vector field \( \mathbf{F} \).
2. **Parametrize the Surface \( S \):**
- Express the surface \( S \) parametrically to accurately relate \( \mathbf{r}(t) \) to the surface.
3. **Determine the Normal Vector \( \mathbf{n} \):**
- The normal vector \( \mathbf{n} \) is assumed to point upwards and should be derived based on the parametrization of the surface.
4. **Compute the Surface Integral:**
- Evaluate the surface integral using the curl of \( \mathbf{F} \) and the unit normal vector \( \mathbf{n} \).
This process should align with the steps outlined in Stokes' Theorem, leading to a thorough understanding and accurate evaluation of the given surface integral.
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