In many problems in the calculus of variations, you need

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