### Solution Found!

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

**Chapter 6, Problem 6.6**

(choose chapter or problem)

**Get Unlimited Answers! Check out our subscriptions**

**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 \(\phi=\phi(\theta)\)

###
Not The Solution You Need? Search for *Your* Answer Here:

### 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 \(\phi=\phi(\theta)\)

**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}\)

### Reviews

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

Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.

No thanks, I don't want to help other students

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

Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.

No thanks, I don't want to help other students

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

Thank you for your recent purchase on StudySoup. We invite you to provide a review below, and help us create a better product.

No thanks, I don't want to help other students