Question

Denoting the three terms of \(\nabla^{2}\) in spherical polars by \(\nabla_{r}^{2}, \nabla_{\theta}^{2}, \nabla_{\phi}^{2}\) in an obvious way, evaluate \(\nabla_{r}^{2} u\), etc. for the two functions given below and verify that, in each case, although the individual terms are not necessarily zero their sum \(\nabla^{2} u\) is zero. Identify the corresponding values of \(\ell\) and \(m\). (a) \(u(r, \theta, \phi)=\left(A r^{2}+\frac{B}{r^{3}}\right) \frac{3 \cos ^{2} \theta-1}{2}\). (b) \(u(r, \theta, \phi)=\left(A r+\frac{B}{r^{2}}\right) \sin \theta \exp i \phi\).

Short answer

Laplacian steps

Step-by-step solution

Understanding the Laplacian in Spherical Coordinates

The Laplacian operator in spherical coordinates is given by: \[ abla^2 u = abla_r^2 u + abla_{\theta}^2 u + abla_{\phi}^2 u \] where: \[ abla_r^2=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial u}{\partial r}\right), \; abla_{\theta}^2=\frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta}\right), \; abla_{\phi}^2=\frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 u}{\partial \phi^2} \]

Calculate \( abla_r^2 u \) for Function (a)

For the function \( u(r, \theta, \phi) = \left(A r^2 + \frac{B}{r^3} \right) \frac{3 \cos^2 \theta - 1}{2} \), we have: \[ \frac{\partial u}{\partial r} = \left( 2 A r - 3 B r^{-4} \right) \frac{3 \cos^2 \theta - 1}{2} \] Next, \[ \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) = \frac{\partial}{\partial r} \left( r^2 \left( 2 A r - 3 B r^{-4} \right) \frac{3 \cos^2 \theta - 1}{2} \right) = \frac{\partial}{\partial r} \left( \left(2 A r^3 - 3 B \right) \frac{3 \cos^2 \theta - 1}{2} \right) \] Simplifying further, \[ \frac{\partial}{\partial r} \left(2 A r^3 - 3 B \right) = 6 A r^2 \] So, \[ abla_r^2 u = \frac{1}{r^2} \cdot 6 A r^2 \frac{3 \cos^2 \theta - 1}{2} = 3 A \left(3 \cos^2 \theta - 1 \right) \]

Calculate \( abla_{\theta}^2 u \) for Function (a)

With the same function: \[ \frac{\partial u}{\partial \theta} = \left(A r^2 + \frac{B}{r^3} \right) 3 \cos \theta (- \sin \theta) \] which simplifies to: \[ \Rightarrow \left(A r^2 + \frac{B}{r^3} \right) 3 \cos \theta (- \sin \theta) \] Next, \[ \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial u}{\partial \theta} \right) = \frac{\partial}{\partial \theta} \left( -3\left(A r^2 + \frac{B}{r^3} \right) \sin^2 \theta \cdot \cos \theta \right) \] Simplifying further, \[ \rightarrow \frac{1}{r^2} \frac{1}{\sin \theta} \left(6 \left(A r^2 + \frac{B}{r^3}\right) \cos \theta \sin \theta + \left(3 Ar^2 + \frac{3B}{r^3}\right)\cos \theta \sin^3 \theta \right) \] gives \[ \rightarrow \left( + 3 \left(A r^2 + \frac{B}{r^3} \right) \cos^2 \theta \right) \] divisable by \frac{2}{r^2 =6 A \cos \theta \sin \theta + 12 \left(A r^2 + \frac{B}{r^3} \right)4 abla_{\theta}^2} \left( \right)

Key concepts

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For instance, they can describe how physical quantities such as temperature, pressure, or electromagnetic fields vary over time and space. PDEs are fundamental in the mathematical modeling of many natural phenomena. Unlike ordinary differential equations, which are defined over a single variable, PDEs involve multiple variables and their partial derivatives. An example of a PDE is the Laplace equation: \[ abla^2 u = 0 \], which is significant in fields like electrostatics, fluid dynamics, and potential theory. Solving PDEs often requires specialized techniques, such as separation of variables.

Spherical Polar Coordinates

Spherical polar coordinates are a three-dimensional coordinate system where the position of a point is specified by three numbers: the radial distance (r), the polar angle (θ), and the azimuthal angle (φ). This system is especially useful in physics and engineering when dealing with problems that exhibit spherical symmetry, such as gravitational fields. The relationships between spherical coordinates \((r, \theta, \theta)\) and Cartesian coordinates \( (x, y, z)\) are:

• \(x = r \, \text{sin}(\theta) \, \text{cos}(\theta)\)
• \(y = r \, \text{sin}(\theta) \, \text{sin}(\theta)\)
• \(z = r \, \text{cos}(\theta)\)

Transforming PDEs into spherical coordinates often simplifies the equations, leveraging their symmetry properties.

Laplacian Operator

The Laplacian operator (∇²) is a differential operator that appears in various PDEs. In spherical coordinates, it decomposes into three parts corresponding to the radial, polar, and azimuthal directions. The formula for the Laplacian in spherical coordinates is: \[abla^2 u = abla_r^2 u + abla_{\theta}^2 u + abla_{\theta}^2 \], where

• \(abla_r^2 = \frac{1}{r^2} \frac{\text{d}}{\text{d}r} \bigg( r^2 \frac{\text{d}u}{\text{d}r} \bigg) \)
• \(abla_{\theta}^2 = \frac{1}{r^2 \text{sin}(\theta) } \frac{\text{d}}{\text{d}\theta} \bigg( \text{sin}(\theta) \frac{\text{d}u}{\text{d}\theta} \bigg) \)
• \(abla_{\theta}^2 = \frac{1}{r^2 \text{sin}^2(\theta) } \frac{\text{d}^2 u}{\text{d}\theta^2} \)

This operator is central to describing phenomena like heat conduction and wave propagation in spherical systems.

Separation of Variables

Separation of variables is a method used to solve certain PDEs, where the equation is rewritten such that each variable can be solved independently. This technique assumes a solution can be written as the product of functions, each depending only on one variable. For instance, a function \(u(r, \theta, \theta) = R(r) P(\theta) P(\theta)\). By substituting this form into a PDE, we can often rearrange the equation such that each term involves only one of the variables. The resulting ordinary differential equations can be solved separately. In spherical coordinates, this method is particularly useful because it simplifies the equations into forms that are more manageable.