Question

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

Short answer

The set of equations representing gradient, divergence, curl and Laplacian in spherical coordinates have been derived using the concept of scale factors.

Step-by-step solution

Understand Scale Factors

In spherical coordinates, the scale factors are the non-dimensional quantities that relate an infinitesimal distance in one direction to the corresponding differential coordinate. We can calculate them by taking the magnitude of the rate of change of the position vector (in this case, spherical radial vector) with respect to each coordinate. The scale factors in spherical coordinates are: \(h_\rho = 1\), \(h_\theta = \rho\), \(h_\phi = \rho \sin \theta\).

Derive Gradient

The gradient is the vector operator that represents the spatial derivative of a scalar function. It has three components in spherical coordinates, each scaled by the scale factor of a given direction. The gradient of a scalar function f can be expressed as: \(\nabla f = \frac{\partial f}{\partial \rho} \hat{{e}}_\rho + \frac{1}{\rho} \frac{\partial f}{\partial \theta} \hat{{e}}_\theta + \frac{1}{\rho \sin \theta} \frac{\partial f}{\partial \phi} \hat{{e}}_\phi \)

Derive Divergence

Divergence is an operation that measures the magnitude of a vector field's source or sink at a given point. It's the dot product of the del operator and a vector field F. The divergence can then be expressed as: \( \nabla \cdot F = \frac{1}{\rho^2} \frac{\partial (\rho^2 F_\rho)}{\partial \rho} + \frac{1}{\rho \sin \theta} \frac{\partial (\sin \theta F_\theta)}{\partial \theta} + \frac{1}{\rho \sin \theta} \frac{\partial F_\phi}{\partial \phi} \)

Derive Curl

Curl is the operation that measures the rotation of a vector field. It's the cross product of del operator and a vector field F. The expression for curl in spherical coordinates can be broken into its three coordinate component parts to give the lengthy formula given in the exercise.

Derive Laplacian

The Laplacian is a differential operator that gauges the divergence of the gradient of a scalar field. In spherical coordinates, it's given by: \( \nabla^2 f = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}(\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial f}{\partial \theta}) + \frac{1}{\rho^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2} \)

Key concepts

Gradient in Spherical Coordinates

The gradient in spherical coordinates provides a way to describe the rate and direction of change of a scalar field. Spherical coordinates consist of three components: radial distance \( \rho \), polar angle \( \theta \), and azimuthal angle \( \phi \). To express the gradient, we use these coordinates to pinpoint any location in three-dimensional space.
The gradient of a scalar function \( f \) in spherical coordinates is defined as:
\[ abla f = \frac{\partial f}{\partial \rho} \hat{\mathbf{e}}_\rho + \frac{1}{\rho} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_\theta + \frac{1}{\rho \sin \theta} \frac{\partial f}{\partial \phi} \hat{\mathbf{e}}_\phi \]
This expression demonstrates how the gradient is composed of directional derivatives. Each component of the gradient indicates how a field changes with a small shift in one of those directions. The scale factors are critical as they adjust the ordinary partial derivatives to the geometric characteristics of spherical coordinates:

• \( h_\rho = 1 \), ensures the radial derivative remains unchanged.
• \( h_\theta = \rho \), adapts the derivative with respect to \( \theta \) due to its angular nature.
• \( h_\phi = \rho \sin \theta \), adjusts the derivative concerning \( \phi \) because of its circumference scaling.

Understanding these geometric influences helps in analytically solving problems in physical contexts like electromagnetism or fluid dynamics.

Divergence in Spherical Coordinates

Divergence measures how much a vector field spreads out from a point, indicating sources or sinks in a flow. In spherical coordinates, divergence of a vector field \( \mathbf{F} \) is calculated by:
\[ abla \cdot \mathbf{F} = \frac{1}{\rho^{2}} \frac{\partial (\rho^{2} F_\rho)}{\partial \rho} + \frac{1}{\rho \sin \theta} \frac{\partial (\sin \theta F_\theta)}{\partial \theta} + \frac{1}{\rho \sin \theta} \frac{\partial F_\phi}{\partial \phi} \]
Each term in this equation manages a part of the vector field’s behavior in spherical space:

• The \( \rho \)-related term evaluates changes in the radial direction, weighted by \( \rho^2 \).
• Angular contributions stem from the \( \theta \) and \( \phi \) terms, correcting for angles by \( \rho \sin \theta \). This accounts for the spherical surface area shrinking or expanding.

This concept is pivotal in physics and engineering to determine the behaviors of fluids and electromagnetic fields as they spread or converge in space.

Curl in Spherical Coordinates

Curl provides a means of quantifying the rotation or twisting of a vector field, represented cross-sectionally. In spherical coordinates, calculating the curl involves derivatives adjusted by spherical geometry scale factors, expressed as:
\[ abla \times \mathbf{F} = \frac{1}{\rho \sin \theta}\left(\frac{\partial(\sin \theta F_{\phi})}{\partial \theta} - \frac{\partial F_{\theta}}{\partial \phi}\right) \hat{\mathbf{e}}_\rho \]\[ + \frac{1}{\rho}\left(\frac{1}{\sin \theta} \frac{\partial F_{\rho}}{\partial \phi} - \frac{\partial(\rho F_{\phi})}{\partial \rho}\right) \hat{\mathbf{e}}_\theta \]\[ + \frac{1}{\rho}\left(\frac{\partial(\rho F_{\theta})}{\partial \rho} - \frac{\partial F_{\rho}}{\partial \theta}\right) \hat{\mathbf{e}}_\phi \]
It’s a detailed expression ensuring each component accounts for spherical topology:

• The \( \rho \)-component handles `\( \theta, \phi \)` interactions.
• The \( \theta \)-component represents the interplay between \( \rho \) and \( \phi \).
• The \( \phi \)-component highlights interactions between \( \rho \) and \( \theta \).

Comprehension of curl is necessary for fields like fluid mechanics or electromagnetism where rotational currents dictate influence.

Laplacian in Spherical Coordinates

The Laplacian operator offers insights into flux convergence and differentials of scalar fields. It combines divergence and gradient operations. In spherical domains, it's vital for tackling wave and heat equations, and its form is:
\[ abla^2 f = \frac{1}{\rho^{2}} \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial f}{\partial \rho}\right) + \frac{1}{\rho^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right) + \frac{1}{\rho^{2} \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^{2}} \]
This equation intricately combines:

• The radial term, highlighting flux changes across expanding-shell volumes.
• The \( \theta \)-derived portion, showcasing flux variations influenced by polar angles.
• The \( \phi \)-dependent factor, considering azimuthal shifts.

Mastering the Laplacian in these contexts is essential for solving issues in fields such as quantum mechanics, optics, or acoustics, where wave behaviors are analyzed.