See photo Findexpression fordtanwhere ū,ū, W an in Ve

Name: See photo Findexpression fordtanwhere ū,ū, W an in Ve
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Description: See photo Findexpression fordtanwhere ū,ū, W an in Ve

Formula: i) ddtut·vt=u't·vt+ut·v't. ii) ddtut×vt=u't×vt+ut×v't Properties of cross product and dot…

Answered: Find expression for dt an where ū, ū, W…

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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Find
expression for
dt
an
where ū,ū, W an in Ve

Expert Solution

Step 1

Formula:

i) $\frac{d}{d t} [u (t) \cdot v (t)] = u' (t) \cdot v (t) + u (t) \cdot v' (t)$ .

ii) $\frac{d}{d t} [u (t) \times v (t)] = u' (t) \times v (t) + u (t) \times v' (t)$

Properties of cross product and dot product:

i) $a \cdot (b \times c) = (a \times b) \cdot c$ .

ii) $a \times b = - b \times a$

Consider $\frac{d}{d t} [u (t) \cdot (v (t) \times w (t))]$ .

$\frac{d}{d t} [u (t) \cdot (v (t) \times w (t))] = u' (t) \cdot [v (t) \times w (t)] + u (t) \cdot \frac{d}{d t} [v (t) \times w (t)] = u' (t) \cdot [v (t) \times w (t)] + u (t) \cdot [v' (t) \times w (t) + v (t) \times w' (t)] = u' (t) \cdot [v (t) \times w (t)] + u (t) \cdot [v' (t) \times w (t)] + u (t) \cdot [v (t) \times w' (t)]$

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