Curvilinear Coordinates Questions and Answers

Answer: D

The unit vector is \(\frac{1}{\sqrt{2}}(\hat{j} - \hat{k})\).

In Cartesian components,

\[ \begin{aligned} \hat{\theta} &= \cos\theta\cos\phi\,\hat{i} + \cos\theta\sin\phi\,\hat{j} \\ &\quad - \sin\theta\,\hat{k} \end{aligned} \]

which does not depend on \(r\). At \(\theta = \pi/4\), \(\phi = \pi/2\): \(\cos\phi = 0\), \(\sin\phi = 1\), and \(\cos\theta = \sin\theta = 1/\sqrt{2}\), so

\[ \hat{\theta} = \tfrac{1}{\sqrt{2}}(\hat{j} - \hat{k}) \]

\(\hat{\theta}\) points in the direction of increasing polar angle, away from the +z axis, so its z-component is negative. \((\hat{j} + \hat{k})/\sqrt{2}\) is \(\hat{r}\) at that point; \((-\hat{j} + \hat{k})/\sqrt{2}\) is \(-\hat{\theta}\); and \(\hat{k}\) is wrong in direction altogether.

Answer: D

The scale factors are \(1, r, r\sin\theta\).

The line element in spherical polar coordinates is

\[ \begin{aligned} ds^{2} &= dr^{2} + r^{2}d\theta^{2} \\ &\quad + r^{2}\sin^{2}\theta\,d\phi^{2} \end{aligned} \]

so the covariant metric is diagonal with \(g_{rr} = 1\), \(g_{\theta\theta} = r^{2}\), \(g_{\phi\phi} = r^{2}\sin^{2}\theta\). The scale (Lamé) factors are the square roots, \(h_{i} = \sqrt{g_{ii}}\): 1, \(r\) and \(r\sin\theta\).

\(1, r^{2}, r^{2}\sin^{2}\theta\) are the metric components themselves, not the scale factors. \(1, r^{2}, \sin^{2}\theta\) is a garbled version of them, and 1, 1, 1 belongs to Cartesian coordinates.

Answer: A

The area element is \(\frac{1}{13}\,d\xi\,d\eta\).

Areas transform with the Jacobian of the change of variables:

\[ \begin{aligned} \frac{\partial(\xi, \eta)}{\partial(x, y)} &= \begin{vmatrix} 2 & 3 \\ 3 & -2 \end{vmatrix} = -13 \\ d\xi\,d\eta &= 13\,dx\,dy \\ dx\,dy &= \tfrac{1}{13}\,d\xi\,d\eta \end{aligned} \]

Only the absolute value of the Jacobian matters for an area. The trap is to use 13 directly instead of its inverse; the options 5 and 3/5 come from adding coefficients instead of taking the determinant.