The surface S is the part of the paraboloid z = 12 – x² – y² that lies above the plane z = oriented upward. 4, If F(z, y, 2) = (- yz, xz, xy) then: (F)-dš = curl %3D (Suggestion: Use the Curl Theorem.) Add Work
The surface S is the part of the paraboloid z = 12 – x² – y² that lies above the plane z = oriented upward. 4, If F(z, y, 2) = (- yz, xz, xy) then: (F)-dš = curl %3D (Suggestion: Use the Curl Theorem.) Add Work
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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![### Problem Statement on Paraboloid and Curl Theorem
The surface \( \mathbf{S} \) is the part of the paraboloid \( z = 12 - x^2 - y^2 \) that lies above the plane \( z = -4 \), oriented upward.
Given the vector field \(\mathbf{F}(x, y, z) = \langle -yz, xz, xy \rangle \), calculate:
\[
\iint_{\mathbf{S}} \text{curl}(\mathbf{F}) \cdot d\mathbf{S} = \,?
\]
(Suggestion: Use the Curl Theorem.)
### Explanation
This problem involves applying the Curl Theorem to evaluate the surface integral involving the curl of a vector field. The setup defines a paraboloid and provides a vector field to work with. The integral expression provided is central to understanding how the curl of the vector field interacts with the defined surface.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdec92e-028d-4f37-af60-841922a4cee5%2F43cc6b96-0985-4dc5-965a-2ba87cb7aa20%2Fi2xn06j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement on Paraboloid and Curl Theorem
The surface \( \mathbf{S} \) is the part of the paraboloid \( z = 12 - x^2 - y^2 \) that lies above the plane \( z = -4 \), oriented upward.
Given the vector field \(\mathbf{F}(x, y, z) = \langle -yz, xz, xy \rangle \), calculate:
\[
\iint_{\mathbf{S}} \text{curl}(\mathbf{F}) \cdot d\mathbf{S} = \,?
\]
(Suggestion: Use the Curl Theorem.)
### Explanation
This problem involves applying the Curl Theorem to evaluate the surface integral involving the curl of a vector field. The setup defines a paraboloid and provides a vector field to work with. The integral expression provided is central to understanding how the curl of the vector field interacts with the defined surface.
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