$$\require{cancel}$$

# 7.E: Coordinates (Exercises)

### Q1

Example 7.4.2 discussed polar coordinates in the Euclidean plane. Use the technique demonstrated in section 7.3 to ﬁnd the metric in these coordinates.

### Q2

Oblique Cartesian coordinates are like normal Cartesian coordinates in the plane, but their axes are at at an angle $$\varphi \neq \pi /2$$ to one another. Show that the metric in these coordinates is

$ds^2 = dx^2 + dy^2 + 2cos\varphi dxdy$

### Q3

Let a $$3$$-plane $$U$$ be deﬁned in Minkowski coordinates by the equation $$x = t$$. Is this plane spacelike, timelike, or lightlike? Find a covector $$S\to$$ that is normal to $$U$$ in the sense described in section 7.6, describing it in terms of its components. Compute the vector $$S$$, also in component form. Verify that $$S\to S = 0$$. Show that $$\to S$$ is tangent to $$M$$.

### Q4

For the oblique Cartesian coordinates deﬁned in problem Q2, use the determinant of the metric to show that the Levi-Civita tensor is

$\epsilon = \begin{pmatrix} 0 & \sin \varphi \\ -\sin \varphi & 0 \end{pmatrix}$

### Q5

Use the technique demonstrated in Example 7.6.6, to ﬁnd the volume of the unit sphere.