Let r → ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) be a path with curvature k ( t ) and define the scaled path r 1 ( t ) = ( λ x ( t ) ) , λ y ( t ) , λ z ( t ) , where λ ≠ 0 is a extant. Prove that curvatures k 1 ( t ) of r 1 ( t ) is k 1 ( t ) = λ − 1 k ( t ) . This explains why the curvature of a circle of Radius R is proportional to 1 R ( in fact it is equal to 1 R ) .

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Answer 1

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Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

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Answer 2

Question

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

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Answer 3

Question

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

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Answer 4

Question

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

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Answer 5

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

Answer 6

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

Answer 7

Chapter 14.4, Problem 82E

To determine

Let $\vec{r} (t) = (x (t), y (t), z (t))$ be a path with curvature $k (t)$ and define the scaled path $r_{1} (t) = (λ x (t)), λ y (t), λ z (t)$ , where $λ \neq 0$ is a extant. Prove that curvatures $k_{1} (t)$ of $r_{1} (t) is k_{1} (t) = λ^{- 1} k (t)$ . This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R} (in fact it is equal to \frac{1}{R})$ .

Answer 8

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Answer 9

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Answer 10

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Answer 11

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

Solution Summary: The author explains how the curvature of a circle of Radius R is proportional to 1 R.

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