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Magazine Chapter 16: Problem 30
Let \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\) \(\begin{array}{l}{\text { Verify each identity. }} \\ {\text { (a) } \nabla \cdot \mathbf{r}=3} \\ {\text { (c) } \nabla^{2} r^{3}=12 r}\end{array} \quad(b) \nabla \cdot(r \mathbf{r})=4 r\)
Short Answer
Expert verified
(a) True, (b) True, (c) True.
Step by step solution
01
Understanding the Notation
The vector \( \mathbf{r} \) is given as \( x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), and \( r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2} \) is the magnitude of the vector \( \mathbf{r} \). Each problem involves verifying a vector calculus identity.
02
Verify \( \nabla \cdot \mathbf{r} = 3 \)
The divergence of a vector \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} \) is given by \( abla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \). Here, \( \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), so \( abla \cdot \mathbf{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3 \). This verifies \( abla \cdot \mathbf{r} = 3 \).
03
Verify \( \nabla \cdot (r \mathbf{r}) = 4r \)
First, write \( r \mathbf{r} = r(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}) = (x r)\mathbf{i} + (y r)\mathbf{j} + (z r)\mathbf{k} \). Compute the divergence: \[ abla \cdot (r \mathbf{r}) = abla \cdot ((xr)\mathbf{i}) + abla \cdot ((yr)\mathbf{j}) + abla \cdot ((zr)\mathbf{k}) \].\Using the product rule: \( abla \cdot (xr) = r \frac{\partial x}{\partial x} + x \frac{\partial r}{\partial x} = r + x \frac{x}{r} = r + \frac{x^2}{r} \).\Similarly, \( abla \cdot (yr) = r + \frac{y^2}{r} \) and \( abla \cdot (zr) = r + \frac{z^2}{r} \).\Combine: \( abla \cdot (r \mathbf{r}) = r + \frac{x^2}{r} + r + \frac{y^2}{r} + r + \frac{z^2}{r} = 3r + \frac{x^2 + y^2 + z^2}{r} = 3r + r = 4r \). This verifies the identity \( abla \cdot (r \mathbf{r}) = 4r \).
04
Verify \( \nabla^2 r^3 = 12r \)
The Laplacian \( abla^2 \) of a scalar function \( f \) in three dimensions is \( abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \). First, find \( r^3 = (x^2 + y^2 + z^2)^{3/2} \). Calculate the Laplacian: 1. Compute \( \frac{\partial r}{\partial x} = \frac{x}{r} \), then \( \frac{\partial r^3}{\partial x} = 3r^2 \cdot \frac{x}{r} = 3xr \). Further \( \frac{\partial^2 r^3}{\partial x^2} = \frac{\partial}{\partial x} (3xr) = 3r + 3x\frac{x}{r} = 3r + 3\frac{x^2}{r} \).2. Similarly, for \( y, z \), we get: \( \frac{\partial^2 r^3}{\partial y^2} = 3r + 3\frac{y^2}{r} \) and \( \frac{\partial^2 r^3}{\partial z^2} = 3r + 3\frac{z^2}{r} \).Summing up: \( abla^2 r^3 = 3r + 3\frac{x^2}{r} + 3r + 3\frac{y^2}{r} + 3r + 3\frac{z^2}{r} = 9r + 3\frac{x^2+y^2+z^2}{r} = 9r + 3r = 12r \). This verifies \( abla^2 r^3 = 12r \).
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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 key component in vector calculus, representing the rate and direction of change in a scalar field. Given a scalar function \(f(x, y, z)\), the gradient \(abla f\) is a vector that points in the direction of the greatest rate of increase of the function.
- The components of the gradient vector are the partial derivatives of the function with respect to each variable.
- Mathematically, if \(f = f(x, y, z)\), then the gradient is expressed as \(abla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\).
Divergence
Divergence is a measure of a vector field's tendency to originate from or converge into a point. It is a scalar value that describes how much the vector field spreads out from a particular location.
- For a vector field \(\mathbf{F} = (F_x, F_y, F_z)\), the divergence is given by \(abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\).
- A positive divergence indicates a 'source' of the field at a point, while a negative value indicates a 'sink'.
Laplacian
The Laplacian operator is a differential operator given by the divergence of the gradient of a function. It is widely used in potential theory, solving equations like the heat equation, or in describing wave propagation.
- In three dimensions, for a scalar function \(f(x, y, z)\), the Laplacian is \(abla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\).
- The Laplacian can reveal information about the concavity or curvature of the function it operates on.
Vector Fields
A vector field assigns a vector to every point in space, providing a way to model various physical phenomena such as gravity, electric fields, and fluid flow.
- In three dimensions, a vector field can be represented as \(\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}\), where \(F_x, F_y,F_z\) are functions of position \(x, y, z\).
- Vector fields are used to visualize how vectors vary across a region and can indicate direction and magnitude at each point.
