Question

For cylindrical coordinates, $$ \begin{aligned} x &=r \cos \theta \\ y &=r \sin \theta \\ z &=z \end{aligned} $$ find the scale factors and derive the following expressions: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial r} \hat{\mathbf{e}}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{\partial f}{\partial z} \hat{\mathbf{e}}_{z} \\ \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial\left(r F_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z} \\ \nabla \times \mathbf{F}=\left(\frac{1}{r} \frac{\partial F_{z}}{\partial \theta}-\frac{\partial F_{\theta}}{\partial z}\right) \hat{\mathbf{e}}_{r}+\left(\frac{\partial F_{r}}{\partial z}-\frac{\partial F_{z}}{\partial r}\right) \hat{\mathbf{e}}_{\theta}+\frac{1}{r}\left(\frac{\partial\left(r F_{\theta}\right)}{\partial r}-\frac{\partial F_{r}}{\partial \theta}\right) \\\ \nabla^{2} f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}. \end{gathered} $$

Short answer

The scale factors in cylindrical coordinates are \(h_r= 1\), \(h_\theta= r\), \(h_z= 1\). The resulting expressions for the gradient, divergence, curl, and Laplacian in cylindrical coordinates correspond to the four given expressions.

Step-by-step solution

Compute the Scale Factors

The scale factors are given by the magnitudes of the derivatives of the position vector with respect to the coordinates. This result in \(h_r= 1\), \(h_\theta= r\), \(h_z= 1\).

Derive Expression for Gradient

The gradient in curvilinear coordinates is given by \(\nabla f=\frac{1}{h_r} \frac{\partial f}{\partial r} \hat{e}_r +\frac{1}{h_\theta} \frac{\partial f}{\partial \theta} \hat{e}_\theta +\frac{1}{h_z} \frac{\partial f}{\partial z} \hat{e}_z\). Substituting the scale factors, we get the first expression.

Derive Expression for Divergence

The divergence in curvilinear coordinates is given by \(\nabla \cdot \mathbf{F}=\frac{1}{h_r h_\theta h_z} \left[ \frac{\partial (h_\theta h_z F_r)}{\partial r} + \frac{\partial (h_r h_z F_\theta )}{\partial \theta} + \frac{\partial (h_r h_\theta F_z)}{\partial z} \right]\). Substituting the scale factors, we get the second expression.

Derive Expression for Curl

The curl in curvilinear coordinates is given by \(\nabla \times \mathbf{F}=\frac{1}{h_r h_\theta h_z} \left[\frac{\partial (h_r F_z)-\partial (h_z F_\theta)}{\partial \theta \partial z} \hat{\mathbf{e}}_r+ \frac{\partial (h_z F_r)-\partial (h_r F_z)}{\partial z \partial r} \hat{\mathbf{e}}_\theta+ \frac{\partial (h_\theta F_r)-\partial (h_r F_\theta)}{\partial r \partial \theta} \hat{\mathbf{e}}_z\right]\). Substituting the scale factors, we get the third expression.

Derive Expression for Laplacian

The Laplacian in curvilinear coordinates is given by \(\nabla^2 f = \frac{1}{h_r h_\theta h_z} \left[ \frac{\partial }{\partial r} \left(h_\theta h_z \frac{\partial f}{\partial r}\right) + \frac{\partial }{\partial \theta} \left(h_z h_r \frac{\partial f}{\partial \theta}\right) + \frac{\partial }{\partial z} \left(h_r h_\theta \frac{\partial f}{\partial z}\right) \right]\). Substituting the scale factors, we get the fourth expression.

Key concepts

Scale Factors in Cylindrical Coordinates

Understanding the concept of scale factors is essential when working with cylindrical coordinates. Scale factors help to relate the metric properties of a curvilinear coordinate system to those of Cartesian coordinates. In cylindrical coordinates, we use the variables \(r, \theta, z\) to describe the position of a point in space in relation to a central axis.

In terms of scale factors, these variables correspond to \(h_r, h_\theta, h_z\), respectively. Each scale factor represents the rate at which the position of a point changes with respect to its corresponding coordinate. For cylindrical coordinates, deriving the scale factors involves taking the derivatives of the position vector with respect to each coordinate variable:

• \(h_r = 1\) for the radial distance,
• \(h_\theta = r\) for the angular component,
• \(h_z = 1\) for the height component.

These scale factors are critical when converting vector operations from Cartesian to cylindrical coordinates. They adjust for the non-uniformity in the measurement of distance, area, or volume elements within this coordinate system.

Gradient in Curvilinear Coordinates

When we calculate the gradient of a function in curvilinear coordinates, such as cylindrical or spherical coordinates, the result is not as straightforward as in Cartesian coordinates. The gradient points in the direction of the steepest ascent of a function and its magnitude represents the rate of increase in that direction.

The general formula for the gradient in curvilinear coordinates involves the scale factors for each dimension. In cylindrical coordinates, the gradient of any scalar function \(f\) is given by:
\[abla f = \frac{1}{h_r} \frac{\partial f}{\partial r} \hat{\mathbf{e}}_r + \frac{1}{h_\theta} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_\theta + \frac{1}{h_z} \frac{\partial f}{\partial z} \hat{\mathbf{e}}_z\]
By inserting the corresponding scale factors \(h_r, h_\theta, h_z\), the expression simplifies to:
\[abla f = \frac{\partial f}{\partial r} \hat{\mathbf{e}}_r + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_\theta + \frac{\partial f}{\partial z} \hat{\mathbf{e}}_z\]
This accounts for the non-linearity and variance in distance that can exist across different directions in cylindrical coordinates.

Divergence in Cylindrical Coordinates

Divergence is another vector operation that requires a curved-coordinates adjustment. It measures the magnitude of a vector field's source or sink at a given point, often visualized as the degree to which fluid might be diverging out of (or converging into) that point.

In cylindrical coordinates, expressing divergence requires careful application of the scale factors due to the curvilinear nature of the system. The divergence of a vector field \(\mathbf{F}\) is defined as:
\[abla \cdot \mathbf{F} = \frac{1}{h_r h_\theta h_z} \left[ \frac{\partial (h_\theta h_z F_r)}{\partial r} + \frac{\partial (h_r h_z F_\theta )}{\partial \theta} + \frac{\partial (h_r h_\theta F_z)}{\partial z} \right]\]
With the scale factors for cylindrical coordinates, this becomes:
\[abla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial (r F_r)}{\partial r} + \frac{1}{r} \frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}\]
This equation captures how the vector field behaves in the radial, angular, and vertical directions and is essential for solving a variety of physical problems in these coordinates.

Curl in Cylindrical Coordinates

Curl in cylindrical coordinates reflects the tendency of field lines to curl or rotate around a point, and it is especially useful in electromagnetism and fluid dynamics. It is calculated somewhat similarly to divergence but with a focus on the rotational attribute of the field.

The curl of a vector field \(\mathbf{F}\) in cylindrical coordinates takes into account the non-uniformity of the system and is given by:
\[abla \times \mathbf{F} = \frac{1}{h_r h_\theta h_z} \left[\begin{array}{c} \frac{\partial (h_r F_z)}{\partial \theta} - \frac{\partial (h_z F_\theta)}{\partial z} \ \frac{\partial (h_z F_r)}{\partial z} - \frac{\partial (h_r F_z)}{\partial r} \ \frac{\partial (h_\theta F_r)}{\partial r} - \frac{\partial (h_r F_\theta)}{\partial \theta} \end{array}\right] \times \left[\begin{array}{c} \hat{\mathbf{e}}_r \ \hat{\mathbf{e}}_\theta \ \hat{\mathbf{e}}_z \end{array}\right]\]
After inserting our known cylindrical scale factors, the formula simplifies to:
\[abla \times \mathbf{F} = \left(\frac{1}{r} \frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z}\right) \hat{\mathbf{e}}_r + \left(\frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r}\right) \hat{\mathbf{e}}_\theta + \frac{1}{r}\left(\frac{\partial (r F_\theta)}{\partial r} - \frac{\partial F_r}{\partial \theta}\right)\hat{\mathbf{e}}_z\]
This result provides a clear understanding of how the vector field rotates in three-dimensional space using cylindrical coordinates.

Laplacian in Curvilinear Coordinates

The Laplacian operator in curvilinear coordinates, such as cylindrical or spherical coordinates, serves a significant role in mathematical physics, especially in areas of field theory and potential theory. It describes the behavior of scalar fields and can be thought of as taking the divergence of the gradient of a function.

In cylindrical coordinates, the Laplacian of a scalar function \(f\) is formulated using the scale factors as:
\[abla^2 f = \frac{1}{h_r h_\theta h_z} \left[ \frac{\partial}{\partial r} \left(h_\theta h_z \frac{\partial f}{\partial r}\right) + \frac{\partial}{\partial \theta} \left(h_z h_r \frac{\partial f}{\partial \theta}\right) + \frac{\partial}{\partial z} \left(h_r h_\theta \frac{\partial f}{\partial z}\right) \right]\]
Substituting in the scale factors for cylindrical coordinates simplifies the expression to the following useful form:
\[abla^2 f = \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial f}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2}\]
This could be interpreted as describing how the variation of a function spreads out from a point in all directions, which is vital when solving equations such as the heat equation or wave equation in cylindrical geometries.