Chapter 11: Q. 64 (page 873)

Let $f (t)$ be a differentiable scalar function and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (f (t) r (t)) = f' (t) r (t) + f (t) r' (t)$ . (This is Theorem 11.11 (b).)

Short Answer

Expert verified

Ans:

\frac{d}{d t} (f (t) r (t)) = \frac{d}{d t} (f (t) ⟨ x (t), y (t), z (t) ⟩) = f^{'} (t) r (t) + f (t) r^{'} (t)

Step by step solution

Step 1. Given information:

$f (t)$ is differentiable scalar function and

$r (t)$ is differentiable vector function.

Step 2. Proving :

Consider

$\frac{d}{d t} (f (t) r (t)) = \frac{d}{d t} (f (t) ⟨ x (t), y (t), z (t) ⟩)$

$= \frac{d}{d t} (⟨ f (t) x (t), f (t) y (t), f (t) z (t) ⟩)$

$= \frac{d}{d t} (f (t) x (t)) i + \frac{d}{d t} (f (t) y (t)) j + \frac{d}{d t} (f (t) z (t)) k$

$= (f^{'} (t) x (t) + f (t) x^{'} (t)) i + (f^{'} (t) y (t) + f (t) y^{'} (t)) j +$ $(f^{'} (t) z (t) + f (t) z^{'} (t)) k$

$= (f^{'} (t) x (t) i + f^{'} (t) y (t) j + f^{'} (t) z (t) k) + (\begin{array}{l} f (t) x^{'} (t) i + f (t) y^{'} (t) j + \\ f (t) z^{'} (t) \end{array}) k$

$= f^{'} (t) (x (t) i + y (t) j + z (t) k) + f (t) (x^{'} (t) o + y^{'} (t) j + z^{'} (t) k)$

$= f^{'} (t) ⟨ x (t), y (t), z (t) ⟩ + f (t) <x^{'} (t), y^{'} (t), z^{'} (t)>$

$= f^{'} (t) r (t) + f (t) r^{'} (t)$

Thus $\frac{d}{d t} (f (t) r (t)) = f^{'} (t) r (t) + f (t) r^{'} (t)$

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Let $f (t)$ be a differentiable scalar function and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (f (t) r (t)) = f' (t) r (t) + f (t) r' (t)$ . (This is Theorem 11.11 (b).)

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Proving :

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Let f(t)be a differentiable scalar function and r(t)be a differentiable vector function. Prove that ddt(f(t)r(t))=f'(t)r(t)+f(t)r'(t). (This is Theorem 11.11 (b).)

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

Calculus

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Let $f (t)$ be a differentiable scalar function and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (f (t) r (t)) = f' (t) r (t) + f (t) r' (t)$ . (This is Theorem 11.11 (b).)