Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=(y+z) \mathbf{i}+(x+z) \mathbf{j}+(x+y) \mathbf{k} $$

Step-by-Step Solution

Answer

The divergence is 0, and the curl is \( \mathbf{0} \).

Step 1: Identify Components of \( \mathbf{F} \)

For the vector field \( \mathbf{F} = (y+z) \mathbf{i} + (x+z) \mathbf{j} + (x+y) \mathbf{k} \), the components are:- \( P = y + z \) (component in \( \mathbf{i} \) direction),- \( Q = x + z \) (component in \( \mathbf{j} \) direction),- \( R = x + y \) (component in \( \mathbf{k} \) direction).

Step 2: Calculate the Divergence of \( \mathbf{F} \)

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by:\[ \text{div } \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]By substituting \( P = y + z, Q = x + z, \) and \( R = x + y \):- \( \frac{\partial P}{\partial x} = 0 \)- \( \frac{\partial Q}{\partial y} = 0 \)- \( \frac{\partial R}{\partial z} = 0 \)Therefore, \( \text{div } \mathbf{F} = 0 + 0 + 0 = 0 \).

Step 3: Apply the Curl Formula

The curl of \( \mathbf{F} \) is given by:\[ \text{curl } \mathbf{F} = abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]Substituting for \( P, Q, R \):- \( \frac{\partial R}{\partial y} = 1 \) and \( \frac{\partial Q}{\partial z} = 1 \)- \( \frac{\partial R}{\partial x} = 1 \) and \( \frac{\partial P}{\partial z} = 1 \)- \( \frac{\partial Q}{\partial x} = 1 \) and \( \frac{\partial P}{\partial y} = 1 \)Thus, \( \text{curl } \mathbf{F} = (1-1) \mathbf{i} - (1-1) \mathbf{j} + (1-1) \mathbf{k} = \mathbf{0} \).

Key Concepts

Divergence in Vector Calculus

The concept of divergence is central to vector calculus and helps us understand how a vector field behaves. Imagine a vector field as representing the flow of a fluid. The divergence of this field at a point gives us an idea of whether the fluid is compressing or expanding at that point. It's like checking if more fluid is entering or leaving a tiny volume around that point.
To calculate divergence, we consider the components of a vector field, usually represented as

• \(P\) in the \( \mathbf{i} \) direction,
• \(Q\) in the \( \mathbf{j} \) direction,
• \(R\) in the \( \mathbf{k} \) direction.

The formula for divergence is:\[\text{div } \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]Each term represents the rate of change of the respective component in its corresponding direction.
This means that when you add them up, you get the total "expansion" at that point. If the divergence is zero, like in our example, it means that there is no expansion or compression at any point; the fluid flow is stable or consistent. This property is useful in physics and engineering, especially in fluid and electromagnetic field analysis.

Understanding Curl

Curl is another important operation in vector calculus. It measures the rotation or swirling strength of a vector field at a point.
If you think of a vector field as a current in a stream or wind, curl tells you how much and in what direction the current is rotating.
The formula for curl is given by:\[\text{curl } \mathbf{F} = abla \times \mathbf{F}\]This can be elaborated with its components as:\[\left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\]

• The \(\mathbf{i}\)-component involves the difference between the rate of change of \(R\) with respect to \(y\) and \(Q\) with respect to \(z\).
• The \(\mathbf{j}\)-component involves the rate of change in \(R\) minus that in \(P\).
• The \(\mathbf{k}\)-component looks at the difference between \(Q\) in \(x\) and \(P\) in \(y\).

In simple terms, these calculations tell us about the twisting of the field.
If the curl of a vector field is zero, it signifies that the field does not exhibit local rotations, making it irrotational, as is the case with our vector field example.

An Introduction to Vector Fields

Vector fields represent quantities with both magnitude and direction throughout space. Think of them like maps which offer directional insights everywhere.
They're used to describe physical entities like gravitational forces, electric fields, and fluid flows. Each point in a vector field is associated with a vector, often depicted as an arrow indicating direction and length for strength.
In its essence, a vector field \(\mathbf{F}(x, y, z)\) is defined by components linked to specific directions (\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)), such as:

• \( P(x,y,z) \) for \( \mathbf{i} \) or the x-direction,
• \( Q(x,y,z) \) for \( \mathbf{j} \),
• \( R(x,y,z) \) for \( \mathbf{k} \).

Understanding vector fields involves looking at how these components interact and change.Manipulations like divergence and curl help interpret the behavior of the field.
For example, they can determine whether a fluid stagnates or if a magnetic field twists. Grasping vector fields and their properties gives insights into complex physical systems, enabling us to solve real-world challenges.