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
Step by step solution
Understand the Divergence Formula
Calculate Divergence for \( \mathbf{v} = k \mathbf{r} \)
Calculate Divergence for \( \mathbf{v} = k(z, x, y) \)
Calculate Divergence for \( \mathbf{v} = k(z, y, x) \)
Calculate Divergence for \( \mathbf{v} = k(x, y, -2z) \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Calculus
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.
Three-Dimensional Vectors
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.
Differential Operators
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.