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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 particle of mass \(m\) moving in two dimensions, subject to a force \(\mathbf{F}=-k x \hat{\mathbf{x}}+K \hat{\mathbf{y}}\) where \(k\) and \(K\) are positive constants. Write down the Hamiltonian and Hamilton's equations, using \(x\) and \(y\) as generalized coordinates. Solve the latter and describe the motion.

The simple form \(\mathcal{H}=T+U\) is true only if your generalized coordinates are "natural" (relation betweeen generalized and underlying Cartesian coordinates is independent of time). If the generalized coordinates are not "natural," you must use the definition \(\mathcal{H}=\sum p_{i} \dot{q}_{i}-\mathcal{L} .\) To illustrate this point, consider the following: Two children are playing catch inside a railroad car that is moving with varying speed \(V\) along a straight horizontal track. For generalized coordinates you can use the position \((x, y, z)\) of the ball relative to a point fixed in the car, but in setting up the Hamiltonian you must use coordinates in an inertial frame \(-\) a frame fixed to the ground. Find the Hamiltonian for the ball and show that it is not equal to \(T+U\) (neither as measured in the car, nor as measured in the ground-based frame).

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.)

Consider a mass \(m\) confined to the \(x\) axis and subject to a force \(F_{x}=k x\) where \(k>0 .\) (a) Write down and sketch the potential energy \(U(x)\) and describe the possible motions of the mass. (Distinguish between the cases that \(E>0 \text { and } E<0 .)\) (b) Write down the Hamiltonian \(\mathcal{H}(x, p)\), and describe the possible phase-space orbits for the two cases \(E>0\) and \(E<0\). (Remember that the function \(\mathcal{H}(x, p)\) must equal the constant energy \(E .\) ) Explain your answers to part (b) in terms of those to part (a).

(a) Evaluate \(\left.\nabla \cdot \mathbf{v} \text { for } \mathbf{v}=k \hat{\mathbf{r}} / r^{2} \text { using rectangular coordinates. (Note that } \hat{\mathbf{r}} / r^{2}=\mathbf{r} / r^{3} .\right)\) (b) Inside the back cover, you will find expressions for the various vector operators (divergence, gradient, etc.) in polar coordinates. Use the expression for the divergence in spherical polar coordinates to confirm your answer to part (a). (Take \(r \neq 0\).)

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