For the first few problems in this set, let *-0 -0 - -) -- F₁ = F₂ F3 y² F₁ = F5 = 22 1. Let S denote the unit sphere cos u sin v sin u sin v COS U T(u, v) = and compute (a) fs F₁ ds (b) ff F2 ds . (c) ff F3 d5 (d) ffs F4 d5 (e) ffs F5-ds 3 (a) ff F₁ d5 . (b) ff F2 ds (c) ff F3 d5 . (d) ffs F₁-dS (e) fs F5-dS . 0 < u < 2π, 050 Σπ Y -X 0 2. Let S denote the square with corners (1, 1, 1), (-1, 1, 1), (1,-1, 1), and (-1,-1, 1), oriented so that the top of S points in the direction of the positive Z-axis, and compute
For the first few problems in this set, let *-0 -0 - -) -- F₁ = F₂ F3 y² F₁ = F5 = 22 1. Let S denote the unit sphere cos u sin v sin u sin v COS U T(u, v) = and compute (a) fs F₁ ds (b) ff F2 ds . (c) ff F3 d5 (d) ffs F4 d5 (e) ffs F5-ds 3 (a) ff F₁ d5 . (b) ff F2 ds (c) ff F3 d5 . (d) ffs F₁-dS (e) fs F5-dS . 0 < u < 2π, 050 Σπ Y -X 0 2. Let S denote the square with corners (1, 1, 1), (-1, 1, 1), (1,-1, 1), and (-1,-1, 1), oriented so that the top of S points in the direction of the positive Z-axis, and compute
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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If you could work number 2 I w
![For the first few problems in this set, let
\[ \mathbf{F}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \quad \mathbf{F}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \quad \mathbf{F}_3 = \begin{pmatrix} x^2 \\ y^2 \\ z^2 \end{pmatrix} \quad \mathbf{F}_4 = \begin{pmatrix} x \\ 0 \\ 0 \end{pmatrix} \quad \mathbf{F}_5 = \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix} \]
1. Let \( S \) denote the unit sphere
\[ \mathbf{T}(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \quad 0 \leq v \leq \pi \]
and compute
(a) \(\iint_{S} \mathbf{F}_1 \cdot d\vec{S}\)
(b) \(\iint_{S} \mathbf{F}_2 \cdot d\vec{S}\)
(c) \(\iint_{S} \mathbf{F}_3 \cdot d\vec{S}\)
(d) \(\iint_{S} \mathbf{F}_4 \cdot d\vec{S}\)
(e) \(\iint_{S} \mathbf{F}_5 \cdot d\vec{S}\)
2. Let \( S \) denote the square with corners \((1, 1, 1), (-1, 1, 1), (1, -1, 1),\) and \((-1, -1, 1)\), oriented so that the top of \( S \) points in the direction of the positive Z-axis, and compute
(a) \(\iint_{S} \mathbf{F}_1 \cdot d\vec{S}\)
(b) \(\iint_{S} \mathbf{F}_2 \cdot d\vec{S}\)
(c) \(\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc158a850-76a9-4504-97b9-8593e0926539%2Fd18f434b-3d28-45d0-9ec1-220ee757e8d0%2Fbmvb4fy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For the first few problems in this set, let
\[ \mathbf{F}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \quad \mathbf{F}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \quad \mathbf{F}_3 = \begin{pmatrix} x^2 \\ y^2 \\ z^2 \end{pmatrix} \quad \mathbf{F}_4 = \begin{pmatrix} x \\ 0 \\ 0 \end{pmatrix} \quad \mathbf{F}_5 = \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix} \]
1. Let \( S \) denote the unit sphere
\[ \mathbf{T}(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \quad 0 \leq v \leq \pi \]
and compute
(a) \(\iint_{S} \mathbf{F}_1 \cdot d\vec{S}\)
(b) \(\iint_{S} \mathbf{F}_2 \cdot d\vec{S}\)
(c) \(\iint_{S} \mathbf{F}_3 \cdot d\vec{S}\)
(d) \(\iint_{S} \mathbf{F}_4 \cdot d\vec{S}\)
(e) \(\iint_{S} \mathbf{F}_5 \cdot d\vec{S}\)
2. Let \( S \) denote the square with corners \((1, 1, 1), (-1, 1, 1), (1, -1, 1),\) and \((-1, -1, 1)\), oriented so that the top of \( S \) points in the direction of the positive Z-axis, and compute
(a) \(\iint_{S} \mathbf{F}_1 \cdot d\vec{S}\)
(b) \(\iint_{S} \mathbf{F}_2 \cdot d\vec{S}\)
(c) \(\
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