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Magazine Chapter 16: Problem 31
Let \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\). Verify each identity. $$\begin{array}{ll}{\text { (a) } \nabla r=\mathbf{r} / r} & {\text { (b) } \nabla \times \mathbf{r}=\mathbf{0}} \\ {\text { (c) } \nabla(1 / r)=-\mathbf{r} / r^{3}} & {\text { (d) } \nabla \ln r=\mathbf{r} / r^{2}}\end{array}$$
Short Answer
Expert verified
Each identity (a)-(d) is verified using the gradient, curl, and derivative rules.
Step by step solution
01
Verify Identity (a)
First, we need to calculate \( abla r \) where \( r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2} \). The gradient of a scalar function \( r \) is given by the partial derivatives: \( abla r = \frac{\partial r}{\partial x} \mathbf{i} + \frac{\partial r}{\partial y} \mathbf{j} + \frac{\partial r}{\partial z} \mathbf{k} \).Calculate each component:- \( \frac{\partial r}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{r} \)- \( \frac{\partial r}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}} = \frac{y}{r} \)- \( \frac{\partial r}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}} = \frac{z}{r} \)Thus, \( abla r = \frac{x}{r} \mathbf{i} + \frac{y}{r} \mathbf{j} + \frac{z}{r} \mathbf{k} = \frac{\mathbf{r}}{r} \), which verifies (a).
02
Verify Identity (b)
To show \( abla \times \mathbf{r} = \mathbf{0} \), consider that \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). The curl of a vector field \( \mathbf{F} = \mathbf{i}F_1 + \mathbf{j}F_2 + \mathbf{k}F_3 \) is given by the determinant:\[abla \times \mathbf{r} = \text{det} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ x & y & z \end{vmatrix}\]Calculate the determinant:1. The \( \mathbf{i} \)-component: \( \left(\frac{\partial z}{\partial y} - \frac{\partial y}{\partial z}\right) = 0 - 0 = 0 \)2. The \( \mathbf{j} \)-component: \( \left(\frac{\partial x}{\partial z} - \frac{\partial z}{\partial x}\right) = 0 - 0 = 0 \)3. The \( \mathbf{k} \)-component: \( \left(\frac{\partial y}{\partial x} - \frac{\partial x}{\partial y}\right) = 0 - 0 = 0 \)Thus, \( abla \times \mathbf{r} = \mathbf{0} \), verifying (b).
03
Verify Identity (c)
The gradient operator applied to \( \frac{1}{r} \) is given by \( abla(\frac{1}{r}) \). Use the identity: \( abla u^{-1} = -u^{-2}abla u \).Let \( u = r = \sqrt{x^2 + y^2 + z^2} \), then:- \( abla u = \frac{\mathbf{r}}{r} \) from Step 1.Plug into the identity:\[abla \left( \frac{1}{r} \right) = -r^{-2} \cdot \frac{\mathbf{r}}{r} = -\frac{\mathbf{r}}{r^3}\]Thus, \( abla \left( \frac{1}{r} \right) = -\frac{\mathbf{r}}{r^3} \), verifying (c).
04
Verify Identity (d)
For \( abla \ln r \), use the identity: \( abla \ln u = \frac{abla u}{u} \).Let \( u = r \), then, using results from Step 1:- \( abla u = \frac{\mathbf{r}}{r} \)Compute:\[abla \ln r = \frac{abla r}{r} = \frac{\frac{\mathbf{r}}{r}}{r} = \frac{\mathbf{r}}{r^2}\]Thus, \( abla \ln r = \frac{\mathbf{r}}{r^2} \), verifying (d).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Gradient
The gradient is a fundamental concept in vector calculus, describing how a scalar field changes. For a scalar function like the distance from the origin, represented by \( r = \sqrt{x^2 + y^2 + z^2} \), the gradient shows the direction and rate of change of that distance. The gradient, denoted as \( abla r \), is a vector. It points in the direction of the greatest rate of increase of the function. In this exercise, you see that the gradient \( abla r \) for \( r \) is \( \frac{\mathbf{r}}{r} \), which simplifies to the radial vector \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) divided by its magnitude. This essentially tells us that the fastest way to increase \( r \) is in the direction of \( \mathbf{r} \) itself. The magnitude of the gradient vector can also be thought of as the 'steepness' at each point in the field.
Curl
Curl is another key vector calculus concept that applies to vector fields, rather than scalar fields. It measures the rotation or 'twisting' present in the field. The curl for a vector field \( \mathbf{F} \) is expressed as \( abla \times \mathbf{F} \). This operation uses determinants to calculate rotational aspects of the vector field. In our example, \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), we find that \( abla \times \mathbf{r} = \mathbf{0} \). This zero value indicates that our vector field is irrotational, meaning there is no rotation at any point in the space. Calculating curl helps to understand fields like magnetic fields and fluid flow, where rotation can be a significant factor.
Divergence
Divergence is an operation applicable to vector fields, providing a scalar output that tells how much a vector field 'spreads out' from a point. Specifically, it measures the magnitude of a field's source or sink at a specific point. For instance, consider \( abla \cdot \mathbf{F} \) for some vector field \( \mathbf{F} \). While we did not compute divergence directly in this exercise, recognizing the concepts from the gradients differentiating relationships, such as \( abla(\frac{1}{r}) = -\frac{\mathbf{r}}{r^3} \) and \( abla \ln r = \frac{\mathbf{r}}{r^2} \), hints at how the field behaves. These represent clear relationships where divergence would play a crucial role, especially in fields like electromagnetism, for analyzing charge density.
Vector Fields
Vector fields assign a vector to every point in space, which is vital in describing fields in physics like electromagnetic fields or fluid flow. In this exercise, the vector field \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \) represents a basic radial field. Each point in the field points away from the origin, scaling with distance. This field is foundational for understanding more complex field behaviors. Studying vector fields involves understanding properties like gradient, curl, and divergence to gain insights on movement and strengths of forces within a given space. By applying such concepts, we can predict the behavior of fields in physical applications. Notice how in the solution, vector field identities help verify properties using calculus operations to demonstrate intrinsic characteristics of fields in space.
