rl F) • n dA = ±|0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Evaluate the integral
\[
\iint_{S} (\text{curl } \mathbf{F}) \cdot \mathbf{n} \, dA
\]
directly for
\[
\mathbf{F} = [z^2, -x^2, 0],
\]
where \( S \) is the rectangle with vertices
\[
(0, 0, 0), (4, 0, 0), (0, 4, 4), (4, 4, 4).
\]
**Note:** Enter the exact answer.
---
**Solution Format:**
\[
\iint_{S} (\text{curl } \mathbf{F}) \cdot \mathbf{n} \, dA = \pm 0 \quad [ \times ]
\]
**Explanation:**
This is a vector calculus problem where you are required to directly evaluate the surface integral of the curl of a vector field \(\mathbf{F}\) over a given surface \(S\). The vertices of the rectangle \(S\) define the plane in 3D space. The task involves computing the curl of the vector field, determining the normal vector \(\mathbf{n}\) to the surface, and then performing the surface integral.
The figure or diagram associated with the problem is not visible, but the integral results in zero, which is indicated as an incorrect choice in the context of the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F002560c4-f50e-4f4a-a33c-b3a98b864e70%2Fda23706d-fd1e-4893-975c-be9e319624cc%2F5on1ix_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the integral
\[
\iint_{S} (\text{curl } \mathbf{F}) \cdot \mathbf{n} \, dA
\]
directly for
\[
\mathbf{F} = [z^2, -x^2, 0],
\]
where \( S \) is the rectangle with vertices
\[
(0, 0, 0), (4, 0, 0), (0, 4, 4), (4, 4, 4).
\]
**Note:** Enter the exact answer.
---
**Solution Format:**
\[
\iint_{S} (\text{curl } \mathbf{F}) \cdot \mathbf{n} \, dA = \pm 0 \quad [ \times ]
\]
**Explanation:**
This is a vector calculus problem where you are required to directly evaluate the surface integral of the curl of a vector field \(\mathbf{F}\) over a given surface \(S\). The vertices of the rectangle \(S\) define the plane in 3D space. The task involves computing the curl of the vector field, determining the normal vector \(\mathbf{n}\) to the surface, and then performing the surface integral.
The figure or diagram associated with the problem is not visible, but the integral results in zero, which is indicated as an incorrect choice in the context of the problem.
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