1. The value of ls F dS for the vector field F=< x'2, 2ryz, az> where S is the surface of the region E that is the rectangular box -1
1. The value of ls F dS for the vector field F=< x'2, 2ryz, az> where S is the surface of the region E that is the rectangular box -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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![**1.**
The value of \(\iiint \vec{F} \cdot d\vec{S}\) for the vector field \(\vec{F} = \langle x^2x^3, 2xyz^3, xz^4 \rangle\) where \(S\) is the surface of the region \(E\) that is the rectangular box \(-1 \leq x \leq 1, -2 \leq y \leq 2, -3 \leq z \leq 3\) is an integer \(A\). What is \(A\)?
---
**2.**
The value of \(\iint_S \vec{F} \cdot d\vec{S}\) for \(\vec{F}(x, y, z) = \langle xy, (y^2 + e^{xz^2}), \sin(xy) \rangle\) and \(S\) is the surface of the region bounded by \(z = 1 - x^2, z = 0, y = 0, y = 2\) is an integer \(A\). What is \(A\)?
---
**3.**
What is true about the value of \(\iint_S F \cdot dS\) for the vector field \(F = \langle x^3y, -x^2y^2, -x^2yz \rangle\) where \(S\) is the surface of the hyperboloid \(x^2 + y^2 - z^2 = 1\) between \(z = -2\) and \(z = 2\)?
- (a) It’s less than 0.
- (b) It’s equal to 0.
- (c) It’s greater than 0.
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**4.**
The value of \(\iint_S F \cdot dS\) for the vector field \(F = \langle x^2y, xy^2, 2xyz \rangle\) where \(S\) is the surface of the tetrahedron formed by the plane \(x + 2y + z = 2\) in the first octant, is
\[
\iint_S F \cdot dS = \frac{A}{B}
\]
where \(A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11cd6b3d-6bf5-4443-bba6-ce8779e6a805%2F25b45545-eaf9-4c2a-a869-924ef9c09643%2F3hl5kie_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**1.**
The value of \(\iiint \vec{F} \cdot d\vec{S}\) for the vector field \(\vec{F} = \langle x^2x^3, 2xyz^3, xz^4 \rangle\) where \(S\) is the surface of the region \(E\) that is the rectangular box \(-1 \leq x \leq 1, -2 \leq y \leq 2, -3 \leq z \leq 3\) is an integer \(A\). What is \(A\)?
---
**2.**
The value of \(\iint_S \vec{F} \cdot d\vec{S}\) for \(\vec{F}(x, y, z) = \langle xy, (y^2 + e^{xz^2}), \sin(xy) \rangle\) and \(S\) is the surface of the region bounded by \(z = 1 - x^2, z = 0, y = 0, y = 2\) is an integer \(A\). What is \(A\)?
---
**3.**
What is true about the value of \(\iint_S F \cdot dS\) for the vector field \(F = \langle x^3y, -x^2y^2, -x^2yz \rangle\) where \(S\) is the surface of the hyperboloid \(x^2 + y^2 - z^2 = 1\) between \(z = -2\) and \(z = 2\)?
- (a) It’s less than 0.
- (b) It’s equal to 0.
- (c) It’s greater than 0.
---
**4.**
The value of \(\iint_S F \cdot dS\) for the vector field \(F = \langle x^2y, xy^2, 2xyz \rangle\) where \(S\) is the surface of the tetrahedron formed by the plane \(x + 2y + z = 2\) in the first octant, is
\[
\iint_S F \cdot dS = \frac{A}{B}
\]
where \(A
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