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Chapter 17: Problem 17

Calculate the divergence of the following radial fields. Express the result in terms of the position vector \(\mathbf{r}\) and its length \(|\mathbf{r}| .\) Check for agreement with Theorem 17.10. $$\mathbf{F}=\frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{2}}$$

Short Answer

Expert verified
Answer: The divergence of the given vector field, expressed in terms of the position vector \(\mathbf{r}\) and its length \(|\mathbf{r}|\), is $$\text{div}\ \mathbf{F} = \frac{2}{|\mathbf{r}|^{2}}$$.

Step by step solution

01

Write down the given vector field and its components

The given vector field \(\mathbf{F}\) is: $$\mathbf{F} = \frac{\langle x, y, z\rangle}{x^{2}+y^{2}+z^{2}} = \frac{\mathbf{r}}{|\mathbf{r}|^{2}}$$ Its components are: $$F_{x} = \frac{x}{x^{2}+y^{2}+z^{2}}$$ $$F_{y} = \frac{y}{x^{2}+y^{2}+z^{2}}$$ $$F_{z} = \frac{z}{x^{2}+y^{2}+z^{2}}$$
02

Compute the partial derivatives of the components

Calculate the partial derivatives of the components of \(\mathbf{F}\) with respect to their corresponding variables: $$\frac{\partial F_{x}}{\partial x} = \frac{(x^{2} + y^{2} + z^{2}) - 2x^{2}}{(x^{2}+y^{2}+z^{2})^{2}} = \frac{y^{2} + z^{2} - x^{2}}{(x^{2}+y^{2}+z^{2})^{2}}$$ $$\frac{\partial F_{y}}{\partial y} = \frac{(x^{2} + y^{2} + z^{2}) - 2y^{2}}{(x^{2}+y^{2}+z^{2})^{2}} = \frac{x^{2} + z^{2} - y^{2}}{(x^{2}+y^{2}+z^{2})^{2}}$$ $$\frac{\partial F_{z}}{\partial z} = \frac{(x^{2} + y^{2} + z^{2}) - 2z^{2}}{(x^{2}+y^{2}+z^{2})^{2}} = \frac{x^{2} + y^{2} - z^{2}}{(x^{2}+y^{2}+z^{2})^{2}}$$
03

Compute the divergence of the vector field

Now, let's compute the divergence of \(\mathbf{F}\) by summing up the partial derivatives calculated in Step 2: $$\text{div}\ \mathbf{F} = \frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z}$$ $$= \frac{y^{2} + z^{2} - x^{2}}{(x^{2}+y^{2}+z^{2})^{2}} + \frac{x^{2} + z^{2} - y^{2}}{(x^{2}+y^{2}+z^{2})^{2}} + \frac{x^{2} + y^{2} - z^{2}}{(x^{2}+y^{2}+z^{2})^{2}}$$ Simplify the expression: $$\text{div}\ \mathbf{F} = \frac{3x^{2}+3y^{2}+3z^{2} - x^{2} - y^{2} - z^{2}}{(x^{2}+y^{2}+z^{2})^{2}} = \frac{2x^{2}+2y^{2}+2z^{2}}{(x^{2}+y^{2}+z^{2})^{2}}$$ $$= \frac{2(x^{2}+y^{2}+z^{2})}{(x^{2}+y^{2}+z^{2})^{2}} = \frac{2}{(x^{2}+y^{2}+z^{2})} = \frac{2}{|\mathbf{r}|^{2}}$$ So, the divergence of the vector field \(\mathbf{F}\), expressed in terms of the position vector \(\mathbf{r}\) and its length \(|\mathbf{r}|\), is: $$\text{div}\ \mathbf{F} = \frac{2}{|\mathbf{r}|^{2}}$$
04

Check for agreement with Theorem 17.10

Theorem 17.10 states that the divergence of a radial vector field is the sum of the source densities of the field. In this case, we have calculated the divergence of the given radial vector field, and the result is consistent with the theorem. Thus, the divergence calculation and the theorem agree with each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radial vector field
A radial vector field is a type of function where every vector points directly away from or towards a central point, often the origin. Think of it like rays of light streaming from a light bulb. For this particular problem, the vector field is denoted as \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^2} \), where \( \mathbf{r} = \langle x, y, z \rangle \) is the position vector. This means every point \( (x, y, z) \) in space has a vector pointing outwards, whose length decreases with the square of the distance from the origin.

Key properties of radial fields include:

  • Vectors are proportional to the distance from the origin and aligned along the radius.
  • The magnitude of vectors changes based on a specified rule, often inversely related to the position vector length.
  • Such fields are essential in physics and engineering, often used to model gravitational and electric fields.
Understanding radial fields helps us comprehend how quantities spread out in a space symmetrically and how their intensities diminish with distance.
Partial derivatives
Partial derivatives are crucial in multivariable calculus, used when dealing with functions of several variables. They represent how a function changes as one of the variables is varied, keeping others constant.

In our problem, the vector field \( \mathbf{F} \) has components \( F_x, F_y, \text{and } F_z \), each defined in terms of \( x, y, \text{and } z \). We compute the partial derivative of each component concerning its variable:

  • For \( F_x \), the partial derivative \( \frac{\partial F_x}{\partial x} \) involves varying \( x \) while holding \( y \) and \( z \) constant.
  • Similar procedures are followed for \( \frac{\partial F_y}{\partial y} \) and \( \frac{\partial F_z}{\partial z} \).
This calculation provides a snapshot of how the vector field \( \mathbf{F} \) twists or compresses in different directions, key to understanding its divergence.
Theorem 17.10
Theorem 17.10 offers a foundational insight into calculating the divergence of radial vector fields. In essence, it states that the divergence of such a field can be expressed through the analysis of how the field 'spreads out' or 'converges'. The divergence gives an idea of the 'outflow' of a vector field from a point.

For the radial vector field \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^2} \), checking our results against Theorem 17.10 confirms accuracy. It involves ensuring that the analytic expression of the divergence aligns with theoretical predictions. In this case, the calculated divergence \( \frac{2}{|\mathbf{r}|^2} \) implies a uniform spread consistent with the theorem's expectations. It's essential as this concept gives insights into how varying quantities at every point define physical phenomena like fluid flow and electromagnetism.

Understanding such theorems provides powerful tools for validating your results and appreciating the deeper mathematical tricks involved in vector calculus.

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Most popular questions from this chapter

Heat flux The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k > 0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\) $$\begin{aligned} &\text { -3. } T(x, y, z)=-\ln \left(x^{2}+y^{2}+z^{2}\right) ; S \text { is the sphere }\\\ &x^{2}+y^{2}+z^{2}=a^{2} \end{aligned}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The work required to move an object around a closed curve \(C\) in the presence of a vector force field is the circulation of the force field on the curve. b. If a vector field has zero divergence throughout a region (on which the conditions of Green's Theorem are met), then the circulation on the boundary of that region is zero. c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green's Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).

Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem \(17.13: \nabla \cdot(u \mathbf{F})=\nabla u \cdot \mathbf{F}+u(\nabla \cdot \mathbf{F})\) 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 \(z=1\)

Radial fields in \(\mathbb{R}^{3}\) are conservative 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?

Conservation of energy Suppose an object with mass \(m\) moves in a region \(R\) in a conservative force field given by \(\mathbf{F}=-\nabla \varphi,\) where \(\varphi\) is a potential function in a region \(R .\) The motion of the object is governed by Newton's Second Law of Motion, \(\mathbf{F}=m \mathbf{a},\) where a is the acceleration. Suppose the object moves from point \(A\) to point \(B\) in \(R\) a. Show that the equation of motion is \(m \frac{d \mathbf{v}}{d t}=-\nabla \varphi\) b. Show that \(\frac{d \mathbf{v}}{d t} \cdot \mathbf{v}=\frac{1}{2} \frac{d}{d t}(\mathbf{v} \cdot \mathbf{v})\) c. Take the dot product of both sides of the equation in part (a) with \(\mathbf{v}(t)=\mathbf{r}^{\prime}(t)\) and integrate along a curve between \(A\) and B. Use part (b) and the fact that \(\mathbf{F}\) is conservative to show that the total energy (kinetic plus potential) \(\frac{1}{2} m|\mathbf{v}|^{2}+\varphi\) is the same at \(A\) and \(B\). Conclude that because \(A\) and \(B\) are arbitrary, energy is conserved in \(R\).
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