Chapter 13: Problem 34
(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
Step by step solution
Express the Vector Field in Rectangular Coordinates
Write the Divergence in Rectangular Coordinates
Calculate the Partial Derivatives
Apply Quotient Rule to Each Component
Simplify Each Term and Sum to Find Divergence
Use Divergence in Spherical Coordinates
Confirm the Result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Field
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
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
- \( r \) - 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
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.