Use Stoke's Theorem to evaluate F. dr where F = ( e¯⁹x – 7yz, e−¹y + 7xz, e−²²) and C is the circle x² + y² = 49 on the plane z = 7 having traversed counterclockwise orientation when viewed from above. The line integral equals Submit Answer Tries 0/30
Use Stoke's Theorem to evaluate F. dr where F = ( e¯⁹x – 7yz, e−¹y + 7xz, e−²²) and C is the circle x² + y² = 49 on the plane z = 7 having traversed counterclockwise orientation when viewed from above. The line integral equals Submit Answer Tries 0/30
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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![### Using Stokes' Theorem to Evaluate a Line Integral
In this exercise, we will use Stokes' Theorem to evaluate the line integral of a vector field **F** over a curve **C**. The vector field **F** and the curve **C** are defined as follows:
#### Vector Field
\[
**F** = \langle e^{-9x} - 7yz, e^{-1y} + 7xz, e^{-2z} \rangle
\]
#### Curve
The curve **C** is the circle defined by the equation:
\[
x^2 + y^2 = 49
\]
This circle lies in the plane \( z = 7 \) and is traversed in a counterclockwise orientation when viewed from above.
#### Problem Statement
We are tasked to evaluate the line integral:
\[
\oint_C **F** \cdot d**r**
\]
**Input Box:**
"The line integral equals" (Followed by an empty input box for the answer)
#### Submission
There is a button labeled "Submit Answer" below the input box with an indicator "Tries 0/30".
### Explanation of Stokes' Theorem
Stokes' Theorem relates a surface integral over a surface **S** to a line integral over the boundary curve **C** of **S**. It is given by:
\[
\oint_C **F** \cdot d**r** = \iint_S (\nabla \times **F**) \cdot d**S**
\]
Where:
- \(\nabla \times **F**\) is the curl of the vector field **F**.
- \(d**S**\) is the vector surface element of the surface **S**.
#### Steps to Solve:
1. Parameterize \( \textbf{C}: x^2 + y^2 = 49 \) in the plane \(z = 7\).
2. Compute the curl of **F**.
3. Set up and evaluate the surface integral using the curl of **F** over the surface bounded by **C**.
This problem is a practical application of vector calculus and requires knowledge of parameterization, computing curls, and evaluating integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2Fdb223ce7-16d5-467e-9087-a061ca8575c1%2Fzwkutsj_processed.png&w=3840&q=75)
Transcribed Image Text:### Using Stokes' Theorem to Evaluate a Line Integral
In this exercise, we will use Stokes' Theorem to evaluate the line integral of a vector field **F** over a curve **C**. The vector field **F** and the curve **C** are defined as follows:
#### Vector Field
\[
**F** = \langle e^{-9x} - 7yz, e^{-1y} + 7xz, e^{-2z} \rangle
\]
#### Curve
The curve **C** is the circle defined by the equation:
\[
x^2 + y^2 = 49
\]
This circle lies in the plane \( z = 7 \) and is traversed in a counterclockwise orientation when viewed from above.
#### Problem Statement
We are tasked to evaluate the line integral:
\[
\oint_C **F** \cdot d**r**
\]
**Input Box:**
"The line integral equals" (Followed by an empty input box for the answer)
#### Submission
There is a button labeled "Submit Answer" below the input box with an indicator "Tries 0/30".
### Explanation of Stokes' Theorem
Stokes' Theorem relates a surface integral over a surface **S** to a line integral over the boundary curve **C** of **S**. It is given by:
\[
\oint_C **F** \cdot d**r** = \iint_S (\nabla \times **F**) \cdot d**S**
\]
Where:
- \(\nabla \times **F**\) is the curl of the vector field **F**.
- \(d**S**\) is the vector surface element of the surface **S**.
#### Steps to Solve:
1. Parameterize \( \textbf{C}: x^2 + y^2 = 49 \) in the plane \(z = 7\).
2. Compute the curl of **F**.
3. Set up and evaluate the surface integral using the curl of **F** over the surface bounded by **C**.
This problem is a practical application of vector calculus and requires knowledge of parameterization, computing curls, and evaluating integrals.
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