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Chapter 13: Problem 31

Evaluate the three-dimensional divergence \(\nabla \cdot\) v for each of the following vectors: (a) \(\mathbf{v}=k \mathbf{r}\) (b) \(\mathbf{v}=k(z, x, y),(\mathbf{c}) \mathbf{v}=k(z, y, x),(\mathbf{d}) \mathbf{v}=k(x, y,-2 z),\) where \(\mathbf{r}=(x, y, z)\) is the usual position vector and \(k\) is a constant.

Short Answer

Expert verified
(a) 3k; (b) 0; (c) 0; (d) 0

Step by step solution

01

Understand the Divergence Formula

The divergence of a vector field \( \mathbf{v} = (v_x, v_y, v_z) \) in three dimensions is given by:\[ abla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} \] This operator is applied to each of the vector fields in the exercise to find their divergence.
02

Calculate Divergence for \( \mathbf{v} = k \mathbf{r} \)

For \( \mathbf{v} = k \mathbf{r} = k(x, y, z) \), the components are \( v_x = kx \), \( v_y = ky \), and \( v_z = kz \). Thus, apply the divergence formula:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kx) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(kz) = k + k + k = 3k \]
03

Calculate Divergence for \( \mathbf{v} = k(z, x, y) \)

For \( \mathbf{v} = k(z, x, y) \), the components are \( v_x = kz \), \( v_y = kx \), and \( v_z = ky \). Apply the divergence formula:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kz) + \frac{\partial}{\partial y}(kx) + \frac{\partial}{\partial z}(ky) = 0 + 0 + 0 = 0 \]
04

Calculate Divergence for \( \mathbf{v} = k(z, y, x) \)

For \( \mathbf{v} = k(z, y, x) \), the components are \( v_x = kz \), \( v_y = ky \), and \( v_z = kx \). Apply the divergence formula:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kz) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(kx) = 0 + 0 + 0 = 0 \]
05

Calculate Divergence for \( \mathbf{v} = k(x, y, -2z) \)

For \( \mathbf{v} = k(x, y, -2z) \), the components are \( v_x = kx \), \( v_y = ky \), and \( v_z = -2kz \). Apply the divergence formula:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kx) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(-2kz) = k + k - 2k = 0 \]

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Key Concepts

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

Vector Calculus
Vector calculus is a branch of mathematics that deals with vector fields and operations on them. It's essential in physics and engineering, allowing us to describe physical quantities like velocity, force, and acceleration in three-dimensional space.

In vector calculus, we use specific operators to perform operations on vector fields. These operators help us analyze and understand different properties and behaviors of vectors in a mathematical way.
  • Gradient: Calculates the rate and direction of change in a scalar field.
  • Divergence: Describes how much a vector field spreads out or converges at a given point.
  • Curl: Measures the rotation of a vector field around a point.
Each of these has unique applications and can be used to solve problems involving flows, fields, and more. Understanding these concepts enhances our ability to model and interpret complex systems.
Three-Dimensional Vectors
Three-dimensional vectors are powerful ways to represent quantities that have both a magnitude and a direction in space. Each vector has three components usually described by the x, y, and z coordinates.

Consider a vector \( \mathbf{v} = (v_x, v_y, v_z) \). The components \( v_x \), \( v_y \), and \( v_z \) represent how the vector aligns along the x, y, and z-axes, respectively. These vectors can be manipulated, added, subtracted, and scaled to describe various phenomena.
  • Position Vectors: Used to describe the location of a point in space.
  • Velocity Vectors: Describe the speed and direction of movement.
  • Force Vectors: Represent how much and in what direction a push or a pull is applied.
Mastering three-dimensional vectors leads to a better understanding of spatial relationships and interactions.
Differential Operators
Differential operators are tools used to differentiate functions and understand how they change. In vector calculus, these operators allow us to analyze vector fields.

The abla\cdot, or divergence operator, is particularly interesting. It helps in assessing how much a field diverges or concentrates by applying the following formula:

\[abla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\]
  • Applications: Used in fluid dynamics to analyze flows and quantify how particles spread out.
  • Physical Meaning: A positive divergence indicates more fluid is leaving a point than entering, while a negative divergence suggests the opposite.
  • Solving Problems: The divergence operator is applied to different vector fields to calculate how they behave spatially.
By understanding differential operators, students can efficiently solve complex problems dealing with changes within vector fields.

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Most popular questions from this chapter

Consider a mass \(m\) constrained to move in a vertical line under the influence of gravity. Using the coordinate \(x\) measured vertically down from a convenient origin \(O\), write down the Lagrangian \(\mathcal{L}\) and find the generalized momentum \(p=\partial \mathcal{L} / \partial \dot{x}\). Find the Hamiltonian \(\mathcal{H}\) as a function of \(x\) and \(p\) and write down Hamilton's equations of motion. (It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)

A beam of particles is moving along an accelerator pipe in the \(z\) direction. The particles are uniformly distributed in a cylindrical volume of length \(L_{\mathrm{o}}\) (in the \(z\) direction) and radius \(R_{\mathrm{o}} .\) The particles have momenta uniformly distributed with \(p_{z}\) in an interval \(p_{\mathrm{o}} \pm \Delta p_{z}\) and the transverse momentum \(p_{\perp}\) inside a circle of radius \(\Delta p_{\perp} .\) To increase the particles' spatial density, the beam is focused by electric and magnetic fields, so that the radius shrinks to a smaller value \(R\). What does Liouville's theorem tell you about the spread in the transverse momentum \(p_{\perp}\) and the subsequent behavior of the radius \(R ?\) (Assume that the focusing does not affect either \(L_{\mathrm{o}}\) or \(\Delta p_{z}\).)

In the Lagrangian formalism, a coordinate \(q_{i}\) is ignorable if \(\partial \mathcal{L} / \partial q_{i}=0 ;\) that is, if \(\mathcal{L}\) is independent of \(q_{i}\). This guarantees that the momentum \(p_{i}\) is constant. In the Hamiltonian approach, we say that \(q_{i}\) is ignorable if \(\mathcal{H}\) is independent of \(q_{i}\), and this too guarantees \(p_{i}\) is constant. These two conditions must be the same, since the result " \(p_{i}=\) const" is the same either way. Prove directly that this is so, as follows: (a) For a system with one degree of freedom, prove that \(\partial \mathcal{H} / \partial q=-\partial \mathcal{L} / \partial q\) starting from the expression (13.14) for the Hamiltonian. This establishes that \(\partial \mathcal{H} / \partial q=0\) if and only if \(\partial \mathcal{L} / \partial q=0 .\) (b) For a system with \(n\) degrees of freedom, prove that \(\partial \mathcal{H} / \partial q_{i}=-\partial \mathcal{L} / \partial q_{i}\) starting from the expression (13.24).

The general proof of the divergence theorem $$\int_{S} \mathbf{n} \cdot \mathbf{v} d A=\int_{V} \nabla \cdot \mathbf{v} d V$$ is fairly complicated and not especially illuminating. However, there are a few special cases where it is reasonably simple and quite instructive. Here is one: Consider a rectangular region bounded by the six planes \(x=X\) and \(X+A, y=Y\) and \(Y+B,\) and \(z=Z\) and \(Z+C,\) with total volume \(V=A B C .\) The surface \(S\) of this region is made up of six rectangles that we can call \(S_{1}\) (in the plane \(x=X\) ), \(S_{2}\) (in the plane \(x=X+A\) ), and so on. The surface integral on the left of (13.63) is then the sum of six integrals, one over each of the rectangles \(S_{1}, S_{2},\) and so forth. (a) Consider the first two of these integrals and show that $$ \int_{S_{1}} \mathbf{n} \cdot \mathbf{v} d A+\int_{S_{2}} \mathbf{n} \cdot \mathbf{v} d A=\int_{Y}^{Y+B} d y \int_{Z}^{Z+C} d z\left[v_{x}(X+A, y, z)-v_{x}(X, y, z)\right] $$ (b) Show that the integrand on the right can be rewritten as an integral of \(\partial v_{x} / \partial x\) over \(x\) running from \(x=X\) to \(x=X+A .(\text { c) Substitute the result of part }(b)\) into part (a), and write down the corresponding results for the two remaining pairs of faces. Add these results to prove the divergence theorem (13.63).

Set up the Hamiltonian and Hamilton's equations for a projectile of mass \(m\), moving in a vertical plane and subject to gravity but no air resistance. Use as your coordinates \(x\) measured horizontally and \(y\) measured vertically up. Comment on each of the four equations of motion.
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