Problem 25 - Calculus Early Transcendentals

Question

Find \(\nabla \cdot(\nabla \times \mathbf{F})\) $$\mathbf{F}(x, y, z)=\sin x \mathbf{i}+\cos (x-y) \mathbf{j}+z \mathbf{k}$$

Step-by-Step Solution

Answer

The divergence of the curl of \( \mathbf{F} \) is zero.

Step 1: Understand the Problem

We need to find the divergence of the curl of a vector field \( \mathbf{F}(x, y, z) = \sin x \mathbf{i} + \cos(x-y) \mathbf{j} + z \mathbf{k} \). This involves two operations: computing the curl of \( \mathbf{F} \) and then finding the divergence of that result.

Step 2: Compute the Curl

The curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by:\[ abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} - \left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \]For our \( \mathbf{F} \), \( P = \sin x \), \( Q = \cos(x-y) \), and \( R = z \). Calculate each component:- \( \frac{\partial R}{\partial y} = 0 \), \( \frac{\partial Q}{\partial z} = 0 \). Thus, the \( \mathbf{i} \)-component is \( 0 - 0 = 0 \).- \( \frac{\partial R}{\partial x} = 0 \), \( \frac{\partial P}{\partial z} = 0 \). Thus, the \( \mathbf{j} \)-component is \( 0 - 0 = 0 \).- \( \frac{\partial Q}{\partial x} = -\sin(x-y) \), \( \frac{\partial P}{\partial y} = 0 \). Thus, the \( \mathbf{k} \)-component is \(-\sin(x-y) - 0 = -\sin(x-y) \).So, \( abla \times \mathbf{F} = 0 \mathbf{i} + 0 \mathbf{j} - \sin(x-y) \mathbf{k} \).

Step 3: Compute the Divergence

The divergence of a vector field \( \mathbf{G} = G_x \mathbf{i} + G_y \mathbf{j} + G_z \mathbf{k} \) is given by:\[ abla \cdot \mathbf{G} = \frac{\partial G_x}{\partial x} + \frac{\partial G_y}{\partial y} + \frac{\partial G_z}{\partial z} \]For \( abla \times \mathbf{F} \), we have \( G_x = 0 \), \( G_y = 0 \), and \( G_z = -\sin(x-y) \). Calculate each partial derivative:- \( \frac{\partial 0}{\partial x} = 0 \)- \( \frac{\partial 0}{\partial y} = 0 \)- \( \frac{\partial (-\sin(x-y))}{\partial z} = 0 \) (since \( \sin(x-y) \) has no \( z \) term)Thus, \( abla \cdot (abla \times \mathbf{F}) = 0 + 0 + 0 = 0 \).

Step 4: Conclusion

Therefore, the divergence of the curl of the vector field \( \mathbf{F} \) is zero: \( abla \cdot (abla \times \mathbf{F}) = 0 \).

Key Concepts

Curl of a Vector Field

To understand the curl of a vector field, it's essential to grasp what a vector field is first. A vector field assigns a vector to every point in space. For example, wind speed at different points in the atmosphere.
In mathematics, the curl operation helps determine the rotation or the "twisting" effect a vector field exhibits at any given point. Specifically, it measures how much and in what fashion the field is curling around a point. The curl is denoted by \( abla \times \mathbf{F} \) and results in another vector.
This vector depends on changes in the original field components, calculated using partial derivatives. For a 3D vector field with components \( P, Q, R \), the curl is a new vector:

• \( \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} \)
• \( \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} \)
• \( \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \)

In the exercise, the curl of \( \mathbf{F} = \sin x \mathbf{i} + \cos(x-y) \mathbf{j} + z \mathbf{k} \) results in \( 0 \mathbf{i} + 0 \mathbf{j} - \sin(x-y) \mathbf{k} \). This calculation highlights no rotation around the \( \mathbf{i} \) and \( \mathbf{j} \) components but a twist around the \( \mathbf{k} \) component.

Divergence of a Vector Field

The divergence of a vector field provides information about how much a field diverges from a point, essentially representing a measure of the field's "spreading out" at that point. When you think about divergence, picture water emerging from a faucet—the more it spreads away from its source, the greater the divergence.
The mathematical operation for divergence is quite direct. For a vector field \( \mathbf{G} = G_x \mathbf{i} + G_y \mathbf{j} + G_z \mathbf{k} \), the divergence is represented as \( abla \cdot \mathbf{G} \), calculated as:

• \( \frac{\partial G_x}{\partial x} \)
• \( \frac{\partial G_y}{\partial y} \)
• \( \frac{\partial G_z}{\partial z} \)

All added together. If the result is zero, the field neither converges nor diverges at that point—it's a neat balance.
In the given exercise, the divergence of the curl results in zero, \( abla \cdot (abla \times \mathbf{F}) = 0 \). This is a common result since the curl, representing circulation, typically lacks divergence.

Vector Field Operations

Vector field operations like the curl and divergence are fundamental in understanding fluid dynamics, electromagnetism, and other physical phenomena. They allow us to analyze fields more deeply than just visually.
To perform these operations effectively:

• Identify the components of your vector field. This is critical, as it frames the rest of your calculations.
• Apply the correct operation formula (curl or divergence) using partial derivatives. This step involves calculus principles that embody the changes across different spatial dimensions.
• Interpret the results to grasp the physical or mathematical implications of the operations. Curl tells you about the field's local rotation, while divergence indicates expansion or compression.

Understanding and applying vector field operations give insights into the intricacies of the fields, like pinpointing rotational flows or spotting expansions, leading to richer analysis and discovery in various branches of science and engineering.