Question

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

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

Step-by-step solution

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.

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:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kx) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(kz) = k + k + k = 3k \]

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:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kz) + \frac{\partial}{\partial y}(kx) + \frac{\partial}{\partial z}(ky) = 0 + 0 + 0 = 0 \]

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:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kz) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(kx) = 0 + 0 + 0 = 0 \]

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:\[ abla \cdot \mathbf{v} = \frac{\partial}{\partial x}(kx) + \frac{\partial}{\partial y}(ky) + \frac{\partial}{\partial z}(-2kz) = k + k - 2k = 0 \]

Key concepts

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.