Could you do 1c and show steps. I would gre For the first few problems in this set, let\[\vec{F}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \vec{F}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad \vec{F}_3 = \begin{pmatrix} x^2 \\ y^2 \\ z^2 \end{pmatrix}, \quad \vec{F}_4 = \begin{pmatrix} x \\ 0 \\ 0 \end{pmatrix}, \quad \vec{F}_5 = \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix}\]1. Let $ S $ denote the unit sphere\[\vec{T}(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \, 0 \leq v \leq \pi\]and compute(a) $ \iint_S \vec{F}_1 \cdot d\vec{S} $(b) $ \iint_S \vec{F}_2 \cdot d\vec{S} $(c) $ \iint_S \vec{F}_3 \cdot d\vec{S} $(d) $ \iint_S \vec{F}_4 \cdot d\vec{S} $(e) $ \iint_S \vec{F}_5 \cdot d\vec{S} $

Given a parametrization of the unit sphere, ST→(u,v)=(cos⁡usin⁡vsin⁡usin⁡vcos⁡v)Partial…

Answered: For the first few problems in this set,…

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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Could you do 1c and show steps. I would gre

For the first few problems in this set, let

\[
\vec{F}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad \vec{F}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad \vec{F}_3 = \begin{pmatrix} x^2 \\ y^2 \\ z^2 \end{pmatrix}, \quad \vec{F}_4 = \begin{pmatrix} x \\ 0 \\ 0 \end{pmatrix}, \quad \vec{F}_5 = \begin{pmatrix} y \\ -x \\ 0 \end{pmatrix}
\]

1. Let $ S $ denote the unit sphere

\[
\vec{T}(u, v) = \begin{pmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{pmatrix}, \quad 0 \leq u \leq 2\pi, \, 0 \leq v \leq \pi
\]

and compute

(a) $ \iint_S \vec{F}_1 \cdot d\vec{S} $

(b) $ \iint_S \vec{F}_2 \cdot d\vec{S} $

(c) $ \iint_S \vec{F}_3 \cdot d\vec{S} $

(d) $ \iint_S \vec{F}_4 \cdot d\vec{S} $

(e) $ \iint_S \vec{F}_5 \cdot d\vec{S} $

Expert Solution

Step 1: Finding unit normal of the surface

Given a parametrization of the unit sphere, S

$\vec{T} (u, v) = (\begin{matrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{matrix})$

Partial derivatives:

$\frac{\partial \vec{T}}{\partial u} = (\begin{matrix} - \sin u \sin v \\ \cos u \sin v \\ 0 \end{matrix}), \frac{\partial \vec{T}}{\partial v} = (\begin{matrix} \cos u \cos v \\ \sin u \cos v \\ - \sin v \end{matrix})$

$\begin{aligned} \frac{\partial \vec{T}}{\partial u} \times \frac{\partial \vec{T}}{\partial v} & = | \begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ - \sin u \sin v & \cos u \sin v & 0 \\ \cos u \cos v & \sin u \cos v & - \sin v \end{array} | \\ = - (\cos u \sin^{2} v) \vec{i} - (\sin u \sin^{2} v) \vec{j} - (\cos v \sin v) \vec{k} \end{aligned}$

$| \frac{\partial \vec{T}}{\partial u} \times \frac{\partial \vec{T}}{\partial v} | = \sqrt{\cos^{2} u \sin^{4} v + \sin^{2} u \sin^{4} v + \sin^{2} v \cos^{2} v} = \sin v$

Unit vector $\vec{n} = - \frac{\frac{\partial \vec{T}}{\partial u} \times \frac{\partial \vec{T}}{\partial v}}{| \frac{\partial \vec{T}}{\partial u} \times \frac{\partial \vec{T}}{\partial v} |} = (\cos u \sin v) \vec{i} + (\sin u \sin v) \vec{j} + (\cos v) \vec{k}$