Explanation Given info. r ( θ ) = f ( θ ) 〈 cos θ , sin θ 〉 Formula used: e ( θ ) = f ( θ ) 〈 α , β 〉 r ' ( θ ) = f ( θ ) d d θ 〈 α , β 〉 + 〈 α , β 〉 f ' ( θ ) Calculation: r ( θ ) = f ( θ ) 〈 cos θ , sin θ 〉 Then the derivative. r ' ( θ ) = f ( θ ) d d θ 〈 cos θ , sin θ 〉 + 〈 cos θ , sin θ 〉 d d θ f ( θ ) = f ( θ ) 〈 − sin θ , cos θ 〉 + 〈 cos θ , sin θ 〉 f ' ( θ ) = 〈 − sin θ f ( θ ) , cos θ f ( θ ) 〉 + 〈 cos θ f ' ( θ ) , sin θ f ' ( θ ) 〉 = 〈 − sin θ f ( θ ) + cos f ' ( θ ) , cos θ f ( θ ) + sin θ f ' ( θ ) 〉 r ( θ ) . r ' ( θ ) = 〈 f ( θ ) cos θ , f ( θ ) sin θ 〉 . 〈 − sin θ f ( θ ) + cos f ' ( θ ) , cos f ( θ ) + sin θ f ' ( θ ) 〉 = 〈 − sin θ . cos θ f 2 ( θ ) + cos 2 θ f ( θ ) f ' ( θ ) + cos θ sin θ f 2 ( θ ) + sin 2 ( θ ) f ( θ ) f ' ( θ ) 〉 = f ( θ ) f ' ( θ ) [ sin 2 θ + cos 2 θ ] = f ( θ ) f ' ( θ ) r → ( θ ) × r → ' ( θ ) = | i ∧ j ∧ k ∧ f ( θ ) cos θ f ( θ ) sin ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 14.2, Problem 66E

To determine

To prove:

$ψ = \frac{r}{\frac{d r}{d θ}} = \frac{f (θ)}{f' (θ)}$

Here $ψ =$ angle between radial and tangent vector.

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Chapter 14.2, Problem 65E

Chapter 14.2, Problem 67E

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Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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To prove: ψ = r d r d θ = f ( θ ) f ' ( θ ) Here ψ = angle between radial and tangent vector .

Solution Summary: The author explains how to prove = r d.

Videos

To prove:

$ψ = \frac{r}{\frac{d r}{d θ}} = \frac{f (θ)}{f' (θ)}$

Here $ψ =$ angle between radial and tangent vector.

Want to see the full answer?

Chapter 14 Solutions

To prove: ψ = r d r d θ = f ( θ ) f ' ( θ ) Here ψ = angle between radial and tangent vector .

Solution Summary: The author explains how to prove = r d.

Videos

To prove: ψ=rdrdθ=f(θ)f'(θ) Here ψ= angle between radial and tangent vector.

Want to see the full answer?

Chapter 14 Solutions

To prove:

$ψ = \frac{r}{\frac{d r}{d θ}} = \frac{f (θ)}{f' (θ)}$

Here $ψ =$ angle between radial and tangent vector.