Problem 5 - A First Course in Continuum Mechanics

Question

Consider the scalar field \(\phi(\boldsymbol{x})=1 /|\boldsymbol{x}|, \boldsymbol{x} \neq \mathbf{0}\), and the vector field \(\boldsymbol{v}(\boldsymbol{x})=\phi(\boldsymbol{x}) \boldsymbol{n}\), where \(\boldsymbol{n}\) is a constant vector. Show that: (a) \(\nabla \phi(\boldsymbol{x})=-\boldsymbol{x} /|\boldsymbol{x}|^{3}\) for all \(\boldsymbol{x} \neq \mathbf{0}\) (b) \(\Delta \phi(\boldsymbol{x})=0\) for all \(\boldsymbol{x} \neq \mathbf{0}\), (c) \(\Delta \boldsymbol{v}(\boldsymbol{x})=\mathbf{0}\) for all \(\boldsymbol{x} \neq \mathbf{0}\). Remark: Scalar and vector fields satisfying \(\Delta \phi=0\) and \(\Delta \boldsymbol{v}=\mathbf{0}\) in a region \(B\) are said to be harmonic in \(B\). In this sense the above fields are harmonic in any region which excludes the origin.

Step-by-Step Solution

Answer

For part (a), \(abla \phi(\boldsymbol{x}) = -\frac{\boldsymbol{x}}{|\boldsymbol{x}|^3}\). For part (b), \(\Delta \phi(\boldsymbol{x}) = 0\). For part (c), \(\Delta \boldsymbol{v}(\boldsymbol{x}) = \mathbf{0}\).

Step 1: Part (a) - Calculate the Gradient of \(\phi(\boldsymbol{x})\)

To find the gradient, \(abla \phi\), of the scalar field \(\phi(\boldsymbol{x}) = 1/|\boldsymbol{x}|\), use the chain rule and the gradient in spherical coordinates since \(\phi\) depends on the magnitude of \(\boldsymbol{x}\), \(r = |\boldsymbol{x}| = \sqrt{x^2 + y^2 + z^2}\). The gradient in Cartesian coordinates is given by \(abla \phi = \left(\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}\right)\). We compute each partial derivative separately and simplify. Through the computation, we find that \(abla\phi(\boldsymbol{x}) = -\frac{\boldsymbol{x}}{|\boldsymbol{x}|^3}\).

Step 2: Part (b) - Calculate the Laplacian of \(\phi(\boldsymbol{x})\)

To find the Laplacian, \(\Delta \phi\), we apply the Laplacian operator to \(\phi(\boldsymbol{x})\). The Laplacian in Cartesian coordinates is \(\Delta \phi = abla \cdot (abla \phi)\), the divergence of the gradient of \(\phi\). Using the result from part (a), apply the divergence operator to \(abla \phi\), and you will find that \(\Delta \phi(\boldsymbol{x}) = 0\) for all \(\boldsymbol{x} eq \mathbf{0}\).

Step 3: Part (c) - Calculate the Laplacian of \(\boldsymbol{v}(\boldsymbol{x})\)

To demonstrate that \(\Delta \boldsymbol{v} = \mathbf{0}\), we need to compute the Laplacian of the vector field \(\boldsymbol{v} = \phi(\boldsymbol{x}) \boldsymbol{n}\), where \(\boldsymbol{n}\) is a constant vector. The Laplacian of a vector field in Cartesian coordinates can be computed component-wise. Since \(\Delta \phi = 0\) as shown in part (b) and \(\boldsymbol{n}\) is constant, the Laplacian of each component of \(\boldsymbol{v}\) is zero, leading to \(\Delta \boldsymbol{v} = \mathbf{0}\).

Key Concepts

Scalar Field

A scalar field is a mathematical construct which assigns a scalar value to every point in a space. The scalar value can represent any physical quantity which does not have direction, such as temperature or pressure. In the given exercise, \(\phi(\boldsymbol{x})=1 /|\boldsymbol{x}|\), is an example of a scalar field as it gives us a scalar value depending on the point's distance from the origin. Understanding scalar fields helps in visualizing how a certain quantity varies over space.

When studying phenomena described by scalar fields, identifying regions where the field has special characteristics, like where it remains constant or where it experiences rapid change, can be crucial in physical and engineering applications.

Vector Field

A vector field, on the other hand, assigns a vector to each point in space, indicating not only magnitude but also direction. The vector field \(\boldsymbol{v}(\boldsymbol{x})=\phi(\boldsymbol{x}) \boldsymbol{n}\) from our exercise, multiplies the scalar field by a constant vector \(\boldsymbol{n}\), which gives each point in space a vector that has both direction (the same as \(\boldsymbol{n}\)) and magnitude (depending on \(\phi(\boldsymbol{x})\)). Vector fields are fundamental in representing various physical quantities such as velocity fields in fluid dynamics, electromagnetic fields, and force fields like gravity.

Gradient

The concept of a gradient is crucial when it comes to fields. The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field and its magnitude is the rate of that increase. In this exercise, part (a) requires calculating the gradient of the scalar field \(\phi(\boldsymbol{x})\). The result \(abla \phi(\boldsymbol{x})=-\boldsymbol{x} /|\boldsymbol{x}|^{3}\) shows the direction and rate at which \(\phi\) increases in space. The negative sign indicates that \(\phi\) decreases as one moves away from the origin.

Laplacian Operator

The Laplacian operator \(\Delta\) is a scalar operator that measures the divergence of the gradient of a field, effectively representing the rate at which the average value of the field around a point is different from the field at that point. In the exercise, part (b) shows that applying the Laplacian to \(\phi(\boldsymbol{x})\) yields zero, which indicates that the scalar field doesn't have any 'sinks' or 'sources' away from the origin and is harmonic in the region that excludes the origin.

Harmonic Fields

Fields for which the Laplacian is zero everywhere within a specific region are known as harmonic fields. These fields are solutions to Laplace's equation and have applications in many areas of physics including potential theory and electromagnetics. The exercise establishes both scalar field \(\phi\) and vector field \(\boldsymbol{v}\) as harmonic in any region excluding the origin, signifying that they are particularly smooth and well-behaved functions within those regions.

Cartesian Coordinates

Cartesian coordinates are a system in mathematics to uniquely determine each point in a plane by a pair of numerical coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The exercise utilises Cartesian coordinates to express both the scalar and vector fields and perform operations like gradients and Laplacian which are standard in this coordinate system due to its convenience in many physical and engineering problems.

Spherical Coordinates

Spherical coordinates, however, are often more suitable for problems involving symmetry about a point, such as the one given in the exercise. This system defines a point in 3D space with three numbers: the radius from the origin, the polar angle from the positive z-axis, and the azimuthal angle from the positive x-axis on the xy-plane. The gradient operation in part (a) is simplified when using spherical coordinates, which take advantage of the radial symmetry of the scalar field \(\phi\).