### Solution Found! # In many problems in the calculus of variations, you need

Chapter 6, Problem 6.6

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QUESTION:

In many problems in the calculus of variations, you need to know the length $$ds$$ of a short segment of a curve on a surface, as in the expression (6.1). Make a table giving the appropriate expressions for $$ds$$ in the following eight situations:

(a) A curve given by $$y=y(x)$$ in a plane

(b) same but $$x=x(y)$$

(c) same but $$r=r(\phi)$$

(d) same but $$\phi=\phi(r)$$

(e) curve given by $$\phi=\phi(z)$$ on a cylinder of radius $$R$$

(f) same but $$z=z(\phi)$$

(g) curve given by $$\theta=\theta(\phi)$$ on a sphere of radius $$R$$

(h) same but $$\phi=\phi(\theta)$$

### Questions & Answers (3 Reviews)

QUESTION:

In many problems in the calculus of variations, you need to know the length $$ds$$ of a short segment of a curve on a surface, as in the expression (6.1). Make a table giving the appropriate expressions for $$ds$$ in the following eight situations:

(a) A curve given by $$y=y(x)$$ in a plane

(b) same but $$x=x(y)$$

(c) same but $$r=r(\phi)$$

(d) same but $$\phi=\phi(r)$$

(e) curve given by $$\phi=\phi(z)$$ on a cylinder of radius $$R$$

(f) same but $$z=z(\phi)$$

(g) curve given by $$\theta=\theta(\phi)$$ on a sphere of radius $$R$$

(h) same but $$\phi=\phi(\theta)$$

Step 1 of 9

Part (a)

The curve is given as,

$$y=y(x)$$

The above equation can be represented as,

\begin{aligned} d y & =\frac{d y}{d x} d x \\ & =y^{\prime} d x \end{aligned}

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### Review this written solution for 102172) viewed: 510 isbn: 9781891389221 | Classical Mechanics - 0 Edition - Chapter 6 - Problem 6.6

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Textbook: Classical Mechanics

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### Review this written solution for 102172) viewed: 510 isbn: 9781891389221 | Classical Mechanics - 0 Edition - Chapter 6 - Problem 6.6

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Textbook: Classical Mechanics

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