Question

(a) Evaluate \(\left.\nabla \cdot \mathbf{v} \text { for } \mathbf{v}=k \hat{\mathbf{r}} / r^{2} \text { using rectangular coordinates. (Note that } \hat{\mathbf{r}} / r^{2}=\mathbf{r} / r^{3} .\right)\) (b) Inside the back cover, you will find expressions for the various vector operators (divergence, gradient, etc.) in polar coordinates. Use the expression for the divergence in spherical polar coordinates to confirm your answer to part (a). (Take \(r \neq 0\).)

Short answer

The divergence of the given vector field is zero.

Step-by-step solution

Express the Vector Field in Rectangular Coordinates

Given the vector field \(\mathbf{v}= k \hat{\mathbf{r}} / r^2 = k \mathbf{r} / r^3\), where \(\mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}\) and \(r = \sqrt{x^2 + y^2 + z^2}\), express \(\mathbf{v}\) as \(\mathbf{v} = \frac{k}{(x^2 + y^2 + z^2)^{3/2}} (x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}})\).

Write the Divergence in Rectangular Coordinates

The divergence of a vector field \(\mathbf{F} = F_x \hat{\mathbf{i}} + F_y \hat{\mathbf{j}} + F_z \hat{\mathbf{k}}\) in rectangular coordinates is given by \(abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\).

Calculate the Partial Derivatives

For \(\mathbf{v} = \frac{kx}{r^3} \hat{\mathbf{i}} + \frac{ky}{r^3} \hat{\mathbf{j}} + \frac{kz}{r^3} \hat{\mathbf{k}}\), compute \(\frac{\partial}{\partial x}\left(\frac{kx}{r^3}\right)\), \(\frac{\partial}{\partial y}\left(\frac{ky}{r^3}\right)\), and \(\frac{\partial}{\partial z}\left(\frac{kz}{r^3}\right)\). This involves using the quotient rule: \(\frac{\partial}{\partial x}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}\).

Apply Quotient Rule to Each Component

Using the quotient rule, \(\frac{\partial}{\partial x}\left(\frac{kx}{r^3}\right) = \frac{k((x^2+y^2+z^2)^{3/2})-kx\cdot3x(\sqrt{x^2+y^2+z^2})}{(x^2+y^2+z^2)^3}\), similarly calculate for y and z components.

Simplify Each Term and Sum to Find Divergence

After computing each partial derivative and simplifying, the terms combine to give \(abla \cdot \mathbf{v} = 0\).

Use Divergence in Spherical Coordinates

In spherical coordinates, the divergence formula is \(abla \cdot \mathbf{v} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 v_r)\). Substitute \(v_r = k/r^2\) and calculate: \(\frac{\partial}{\partial r}(r^2 \cdot k/r^2) = \frac{\partial}{\partial r}(k) = 0\).

Confirm the Result

Both rectangular and spherical coordinate approaches yield \(abla \cdot \mathbf{v} = 0\), confirming the result.

Key concepts

Vector Field

A vector field is a mathematical construct that assigns a vector to every point in space. This can be visualized as a field of arrows, where each arrow has both a direction and a magnitude. Vector fields are commonly used in physics and engineering to represent things like electromagnetic fields or fluid flows. In mathematics, they are essential for understanding concepts like divergence and curl.
In this exercise, we work with a specific vector field given by \( \mathbf{v} = \frac{k \mathbf{r}}{r^3} \), where \( \mathbf{r} \) is a position vector in space. Vector fields can be expressed in different coordinate systems, such as rectangular or spherical coordinates, each of which can offer unique insights and simplify calculations depending on the symmetry of the situation.

Rectangular Coordinates

Rectangular coordinates are perhaps the most familiar coordinate system, often referred to as Cartesian coordinates. They describe a point in space by three numbers \((x, y, z)\) that correspond to the point's distances along three perpendicular axes. This system is particularly useful because it simplifies the mathematics of many problems, making it easier to apply operations like divergence and curl.
To express our vector field \( \mathbf{v} = \frac{k \mathbf{r}}{r^3} \) in rectangular coordinates, we use \( \mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}} \) and \( r = \sqrt{x^2 + y^2 + z^2} \). This results in the field \( \mathbf{v} = \frac{k}{(x^2 + y^2 + z^2)^{3/2}} (x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}) \).
Rectangular coordinates allow us to calculate the divergence directly by finding the partial derivatives of each component of the vector field and summing them.

Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system that is especially useful when dealing with problems exhibiting spherical symmetry. In spherical coordinates, any point in space is described by three parameters:

• \( r \) - the radial distance from the origin,
• \( \theta \) - the polar angle measured from the positive z-axis,
• \( \phi \) - the azimuthal angle measured from the positive x-axis in the x-y plane.

The divergence of a vector field \( \mathbf{v} \) in spherical coordinates is given by the formula \( abla \cdot \mathbf{v} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 v_r) \), where \( v_r \) is the radial component of the vector field.
In our exercise, substituting \( v_r = \frac{k}{r^2} \) into this formula and performing the differentiation confirms that \( abla \cdot \mathbf{v} = 0 \). This method confirms the result obtained from rectangular coordinates and highlights the power and compatibility of different coordinate systems.

Quotient Rule

The quotient rule is a derivative rule used when differentiating a function that is the quotient of two other functions. It states that if you have a function \( u(x)/v(x) \), its derivative \( \frac{d}{dx}(\frac{u(x)}{v(x)}) \) is given by \[ \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \]This rule is an essential tool when calculating the divergence of vector fields, especially when one or more components involve quotients as is the case in this exercise.
In the exercise, the quotient rule is used to find the derivatives of the components \( \frac{kx}{r^3} \), \( \frac{ky}{r^3} \), and \( \frac{kz}{r^3} \) before summing them to find the overall divergence in rectangular coordinates. Although it may seem complex, using the quotient rule carefully allows you to simplify even the most intricate expressions and derive meaningful conclusions about the vector field.