Question

Express the three-dimensional Dirac delta function as a product of three one- dimensional delta functions involving the coordinates in (a) cylindrical coordinates, (b) \(\quad\) spherical coordinates, (c) general curvilinear coordinates.

Short answer

The three-dimensional Dirac delta function can be expressed as a product of three one-dimensional delta functions as follows: \n(a) For cylindrical coordinates: \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\phi - \phi') \delta(z - z') \), \n(b) For spherical coordinates: \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\theta - \theta') \delta(\phi - \phi') \), \n(c) For general curvilinear coordinates: \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(q^1 - q'^1) \delta(q^2 - q'^2) \delta(q^3 - q'^3) \).

Step-by-step solution

Express Dirac delta in cylindrical coordinates

Let's start with cylindrical coordinates, which are given as \((r, \phi, z)\). The Dirac delta in three dimensions can be expressed as the product of one dimensional delta functions - \( \delta(r - r') \delta(\phi - \phi') \delta(z - z') \). So, in cylindrical coordinates, the three-dimensional Dirac delta function can be written as \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\phi - \phi') \delta(z - z') \).

Express Dirac delta in spherical coordinates

The spherical coordinates are represented as \((r, \theta, \phi)\). In spherical coordinates, the Dirac delta function can again be represented as the product of three one-dimensional delta functions - \( \delta(r - r') \delta(\theta - \theta') \delta(\phi - \phi') \). Thus, in spherical coordinate, it can be written as \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\theta - \theta') \delta(\phi - \phi') \).

Express Dirac delta in general curvilinear coordinates

For general curvilinear coordinates, the delta function can also be expressed as a product of one-dimensional delta functions. If the curvilinear coordinates are \( q^1, q^2, q^3 \), then the Dirac delta function can be expressed as \( \delta(\mathbf{r} - \mathbf{r'}) = \delta(q^1 - q'^1) \delta(q^2 - q'^2) \delta(q^3 - q'^3) \).

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a natural extension of two-dimensional polar coordinates to three dimensions. In this system, a point in space is represented by three components: radial distance \( r \), angular coordinate \( \phi \), and height \( z \).

• Radial distance \( r \): This is the distance from the point to the \( z \)-axis.
• Angular coordinate \( \phi \): This angle is measured in the \( xy \)-plane from the positive \( x \)-axis.
• Height \( z \): This is the point’s height above the \( xy \)-plane.

The Dirac delta function in cylindrical coordinates can be thought of as a product of three one-dimensional delta functions:
\[ \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\phi - \phi') \delta(z - z') \]
This formula simply states that the delta function spikes—signifying a concentrated point—when all three cylindrical coordinates match precisely with a given point \( \mathbf{r'} \).
An important thing to remember is that the interpretation of coordinates can vary based on specific applications and transformations. These variables play a crucial role in fields such as electromagnetism and fluid dynamics.

Spherical Coordinates

Spherical coordinates provide another way of expressing a point in three-dimensional space, often used in scenarios involving spheres. This system comprises the radius \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \).

• Radius \( r \): Represents the distance from the origin to the point.
• Polar Angle \( \theta \): The angle between the line connecting the origin to the point and the positive \( z \)-axis.
• Azimuthal angle \( \phi \): Like in cylindrical coordinates, this is the angle from the positive \( x \)-axis measured in the \( xy \)-plane.

The Dirac delta in spherical coordinates can also be seen as a combination of one-dimensional delta functions:
\[ \delta(\mathbf{r} - \mathbf{r'}) = \delta(r - r') \delta(\theta - \theta') \delta(\phi - \phi') \]
This expression implies the delta function is non-zero only when the spherical coordinates of a point coincide with those specified by \( \mathbf{r'} \).
Spherical coordinates are especially useful in physics problems, such as those involving gravitational and electromagnetic fields around spherical objects.

Curvilinear Coordinates

Curvilinear coordinates are a generalized system of coordinates where the coordinate lines can be curved. They include systems like cylindrical and spherical coordinates, but can be even more general depending on the specific curves adopted.

• General Form: Denoted by \( q^1, q^2, q^3 \), each coordinate can follow a non-linear path.
• Flexibility: This system is adaptable, making it valuable for complex shapes and volumes.

The Dirac delta function in curvilinear coordinates becomes:
\[ \delta(\mathbf{r} - \mathbf{r'}) = \delta(q^1 - q'^1) \delta(q^2 - q'^2) \delta(q^3 - q'^3) \]
This signifies how the Dirac delta function sharpens exactly where the coordinates \( q^i \) match their primed counterparts \( q'^i \).
Curvilinear coordinates are prominently used in advanced calculus and physics for simplifying the handling of geometric complexities in mathematical models.