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In many problems in the calculus of variations, you need
Chapter 6, Problem 6.6(choose chapter or problem)
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 \(\phi=\phi(\theta)\)
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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 \(\phi=\phi(\theta)\)
ANSWER: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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Review this written solution for 102172) viewed: 510 isbn: 9781891389221 | Classical Mechanics - 0 Edition - Chapter 6 - Problem 6.6
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No thanks, I don't want to help other students