Chapter 11: Q. 68 (page 873)

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

Short Answer

Expert verified

$\frac{d}{d t} (‖ r ‖^{2}) = \frac{d}{d t} (r \cdot r) \Rightarrow 2 ‖ r ‖ \frac{d}{d t} (‖ r ‖) = r \cdot \frac{d}{d t} (r) + r \frac{d}{d t} (r) = \frac{1}{‖ r ‖} (r \cdot r^{'})$ Ans:

Step by step solution

Step 1. GIven information:

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

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

To prove this, we use $‖ r ‖^{2} = r \cdot r$

Consider $‖ r ‖^{2} = r \cdot r$

Step 2. Proving:

Differentiating both sides, with respect to 't'

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

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

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

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

Short Answer

Step by step solution

Step 1. GIven information:

Step 2. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

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Geometry

Study anywhere. Anytime. Across all devices.

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