To show that the cylindrical coordinate system is an orthogonal curvilinear coordinate system, we need to verify two key conditions:

1. Curvilinear: The coordinate lines are not straight lines (they curve along the surface of a cylinder), meaning the system is defined by curved coordinate surfaces.
2. Orthogonal: The coordinate lines intersect at right angles (i.e., the coordinate surfaces are perpendicular to each other).

Cylindrical Coordinates Overview

In cylindrical coordinates, a point $P$ in 3D space is represented by the triplet $(r, \theta, z)$, where:

• $r$ is the radial distance from the $z$-axis (distance to the origin in the $xy$-plane).
• $\theta$ is the angle of the point in the $xy$-plane from the positive $x$-axis (azimuthal angle).
• $z$ is the height above the $xy$-plane (same as the Cartesian coordinate).

Step 1: Verify the Curvilinearity

In cylindrical coordinates, the coordinate surfaces are described as follows:

• The surface of constant $r$ is a cylinder with radius $r$ centered on the $z$-axis.
• The surface of constant $\theta$ is a vertical plane passing through the $z$-axis, rotating around the cylinder.
• The surface of constant $z$ is a horizontal plane parallel to the $xy$-plane.

Thus, these surfaces are clearly curved in space and not straight. This confirms the curvilinearity of the coordinate system.

Step 2: Verify the Orthogonality

To check the orthogonality, we need to show that the unit vectors in the directions of increasing $r$, $\theta$, and $z$ are mutually perpendicular.

The position vector $\vec{r}$ in cylindrical coordinates is given by: $\vec{r} = r \hat{r} + z \hat{z}$ where:

• $\hat{r}$ is the unit vector in the direction of increasing $r$ (radial direction),
• $\hat{\theta}$ is the unit vector in the direction of increasing $\theta$ (azimuthal direction),
• $\hat{z}$ is the unit vector in the direction of increasing $z$ (vertical direction).

Calculation of Partial Derivatives

The metric tensor components can be obtained by computing the partial derivatives of the position vector with respect to the coordinates $r$, $\theta$, and $z$. The coordinate basis vectors $\hat{r}$, $\hat{\theta}$, and $\hat{z}$ are given by:

1. For $r$: $\frac{\partial \vec{r}}{\partial r} = \hat{r}$
2. For $\theta$: $\frac{\partial \vec{r}}{\partial \theta} = r \hat{\theta}$
3. For $z$: $\frac{\partial \vec{r}}{\partial z} = \hat{z}$

Next, we check the dot products of these vectors:

1. Dot product of $\frac{\partial \vec{r}}{\partial r}$ and $\frac{\partial \vec{r}}{\partial \theta}$: $\hat{r} \cdot \hat{\theta} = 0$ This shows that $\hat{r}$ is perpendicular to $\hat{\theta}$.

2. Dot product of $\frac{\partial \vec{r}}{\partial r}$ and $\frac{\partial \vec{r}}{\partial z}$: $\hat{r} \cdot \hat{z} = 0$ This shows that $\hat{r}$ is perpendicular to $\hat{z}$.

3. **Dot product of ( \frac{\partial \vec{r}}{\partial