Q. 68 Let r(t) be a differentiable vec... [FREE SOLUTION]

Chapter 11: Q. 68 (page 873)

Let $r (t)$ be a differentiable vector function. Prove that role="math" localid="1649602115972" $\frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} r 路 r^{r} 路$ (Hint: role="math" localid="1649602160237" $鈥� r {鈥�}^{2} = r 路 r)$

Short Answer

Expert verified

$\frac{d}{d t} (鈥� r {鈥�}^{2}) = \frac{d}{d t} (r 路 r) 鈬� 2 鈥� r 鈥� \frac{d}{d t} (鈥� r 鈥�) = r 路 \frac{d}{d t} (r) + r \frac{d}{d t} (r) = \frac{1}{鈥� r 鈥�} (r 路 r^{'})$ Ans:

Step by step solution

Step 1. GIven information:

$r (t)$ is a differentiable vector function.

The objective is to prove that localid="1649602256109" $\frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} r 路 r^{'}$

To prove this, we use $鈥� r {鈥�}^{2} = r 路 r$

Consider $鈥� r {鈥�}^{2} = r 路 r$

Step 2. Proving:

Differentiating both sides, with respect to 't'

$\frac{d}{d t} (鈥� r {鈥�}^{2}) = \frac{d}{d t} (r 路 r) 鈬� 2 鈥� r 鈥� \frac{d}{d t} (鈥� r 鈥�) = r 路 \frac{d}{d t} (r) + r \frac{d}{d t} (r) formula for \frac{d}{d t} (w v) 2 鈥� r 鈥� \frac{d}{d t} (鈥� r 鈥�) = r 路 r^{'} + r 路 r^{'} \frac{d}{d t} (r) = r^{'} 鈬� 2 鈥� r 鈥� \frac{d}{d t} (鈥� r 鈥�) = 2 r 路 r^{'} a d d 鈬� 鈥� r 鈥� \frac{d}{d t} (鈥� r 鈥�) = r 路 r 路 c a n c e l 2 o n both sides 鈬� \frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} (r 路 r^{'}) Divide both sides with 鈥� r 鈥� T h u s \frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} r 路 r^{'} Note : if r (t) i s a differentiable vector function such that 鈥� r (t) 鈥� i s constant then r (t) 路 r^{'} (t) = 0$

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91影视

Let $r (t)$ be a differentiable vector function. Prove that role="math" localid="1649602115972" $\frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} r 路 r^{r} 路$ (Hint: role="math" localid="1649602160237" $鈥� r {鈥�}^{2} = r 路 r)$

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

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91影视

Let r(t) be a differentiable vector function. Prove that role="math" localid="1649602115972" ddt鈥�r鈥�=1鈥�r鈥�r路rr路 (Hint: role="math" localid="1649602160237" 鈥�r鈥�2=r路r

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Probability and Statistics

Pure Maths

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Let $r (t)$ be a differentiable vector function. Prove that role="math" localid="1649602115972" $\frac{d}{d t} (鈥� r 鈥�) = \frac{1}{鈥� r 鈥�} r 路 r^{r} 路$ (Hint: role="math" localid="1649602160237" $鈥� r {鈥�}^{2} = r 路 r)$