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: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial \rho} \hat{\mathbf{e}}_{\rho}+\frac{1}{\rho} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{1}{\rho \sin \theta} \frac{\partial f}{\partial \phi} \hat{\mathbf{e}}_{\phi} \\ \nabla \cdot \mathbf{F}=\frac{1}{\rho^{2}} \frac{\partial\left(\rho^{2} F_{\rho}\right)}{\partial \rho}+\frac{1}{\rho \sin \theta} \frac{\partial\left(\sin \theta F_{\theta}\right)}{\partial \theta}+\frac{1}{\rho \sin \theta} \frac{\partial F_{\phi}}{\partial \phi}. \end{gathered} $$ $$ \begin{aligned} \nabla \times \mathbf{F}=& \frac{1}{\rho \sin \theta}\left(\frac{\partial\left(\sin \theta F_{\phi}\right)}{\partial \theta}-\frac{\partial F_{\theta}}{\partial \phi}\right) \hat{\mathbf{e}}_{\rho}+\frac{1}{\rho}\left(\frac{1}{\sin \theta} \frac{\partial F_{\rho}}{\partial \phi}-\frac{\partial\left(\rho F_{\phi}\right)}{\partial \rho}\right) \\ &+\frac{1}{\rho}\left(\frac{\partial\left(\rho F_{\theta}\right)}{\partial \rho}-\frac{\partial F_{\rho}}{\partial \theta}\right) \hat{\mathbf{e}}_{\phi} \end{aligned} $$ $$ \nabla^{2} f=\frac{1}{\rho^{2}} \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial f}{\partial \rho}\right)+\frac{1}{\rho^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{\rho^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}}. $$

Step by step solution

01

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: \(h_\rho = 1\), \(h_\theta = \rho\), \(h_\phi = \rho \sin \theta\).

02

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: \(\nabla f = \frac{\partial f}{\partial \rho} \hat{{e}}_\rho + \frac{1}{\rho} \frac{\partial f}{\partial \theta} \hat{{e}}_\theta + \frac{1}{\rho \sin \theta} \frac{\partial f}{\partial \phi} \hat{{e}}_\phi \)

03

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: \( \nabla \cdot F = \frac{1}{\rho^2} \frac{\partial (\rho^2 F_\rho)}{\partial \rho} + \frac{1}{\rho \sin \theta} \frac{\partial (\sin \theta F_\theta)}{\partial \theta} + \frac{1}{\rho \sin \theta} \frac{\partial F_\phi}{\partial \phi} \)

04

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.

05

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: \( \nabla^2 f = \frac{1}{\rho^2} \frac{\partial}{\partial \rho}(\rho^2 \frac{\partial f}{\partial \rho}) + \frac{1}{\rho^2 \sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial f}{\partial \theta}) + \frac{1}{\rho^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2} \)

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