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Chapter 14: Problem 6

What is the divergence of an inverse square vector field?

Short Answer

Expert verified
Answer: The divergence of an inverse square vector field is zero.

Step by step solution

01

1. Define the inverse square vector field

An inverse square vector field is given by the expression: \(\vec{F}(x, y, z) = \frac{A}{r^3}\vec{r}\), where \(A\) is a constant, \(r = \sqrt{x^2 + y^2 + z^2}\) is the distance from the origin, and \(\vec{r} = (x, y, z)\) is the position vector.
02

2. Write down the general formula for the divergence of a vector field

The divergence of a vector field \(\vec{F}\), denoted as \(\nabla \cdot \vec{F}\), is given by the sum of the partial derivatives of each component of the field: \(\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\).
03

3. Write down the components of the inverse square vector field

The components of the inverse square vector field are given by: \(F_x = \frac{A}{r^3}x\), \(F_y = \frac{A}{r^3}y\), \(F_z = \frac{A}{r^3}z\).
04

4. Compute the partial derivatives of the components of the vector field

We compute the partial derivatives of the components of the vector field with respect to \(x\), \(y\), and \(z\): \(\frac{\partial F_x}{\partial x} = \frac{A(3x^2 - r^2)}{r^5}\), \(\frac{\partial F_y}{\partial y} = \frac{A(3y^2 - r^2)}{r^5}\), \(\frac{\partial F_z}{\partial z} = \frac{A(3z^2 - r^2)}{r^5}\).
05

5. Compute the divergence of the inverse square vector field

We substitute the computed partial derivatives into the expression for the divergence and sum them: \(\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} = \frac{A(3x^2 - r^2)}{r^5} + \frac{A(3y^2 - r^2)}{r^5} + \frac{A(3z^2 - r^2)}{r^5}\). Simplifying the expression, we get: \(\nabla \cdot \vec{F} = \frac{A(3x^2 + 3y^2 + 3z^2 - 3r^2)}{r^5} = \frac{A(3r^2 - 3r^2)}{r^5} = 0\). Thus, the divergence of an inverse square vector field is zero.

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Most popular questions from this chapter

Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields $$\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i} \quad \mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}$$ satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

Show that \(\left|\mathbf{t}_{u} \times \mathbf{t}_{v}\right|=a^{2} \sin u\) for a sphere of radius \(a\) defined parametrically by \(\mathbf{r}(u, v)=\langle a \sin u \cos v, a \sin u \sin v, a \cos u\rangle,\) where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\).

Let \(\mathbf{F}=\langle z, 0,-y\rangle\). a. What is the component of curl \(\mathbf{F}\) in the direction \(\mathbf{n}=\langle 1,0,0\rangle ?\) b. What is the component of curl \(\mathbf{F}\) in the direction \(\mathbf{n}=\langle 1,-1,1\rangle ?\) c. In what direction \(\mathbf{n}\) is the dot product (curl \(\mathbf{F}\) ) \(\cdot \mathbf{n}\) a maximum?

Recall the Product Rule of Theorem \(14.11: \nabla \cdot(u \mathbf{F})=\nabla u \cdot \mathbf{F}+u(\nabla \cdot \mathbf{F})\) a. Integrate both sides of this identity over a solid region \(D\) with a closed boundary \(S\) and use the Divergence Theorem to prove an integration by parts rule: $$\iiint_{D} u(\nabla \cdot \mathbf{F}) d V=\iint_{S} u \mathbf{F} \cdot \mathbf{n} d S-\iiint_{D} \nabla u \cdot \mathbf{F} d V$$ b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate \(\iiint_{D}\left(x^{2} y+y^{2} z+z^{2} x\right) d V\) where \(D\) is the cube in the first octant cut by the planes \(x=1\) \(y=1,\) and \(z=1\)
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