Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider the velocity field \(\mathbf{v}(x, y, z)=-\omega y \mathbf{i}+\omega x \mathbf{j}\), \(\omega>0\) (see Example 2 and Figure 1). Note that \(\mathbf{v}\) is perpendicular to \(x \mathbf{i}+y \mathbf{j}\) and that \(\|\mathbf{v}\|=\omega \sqrt{x^{2}+y^{2}}\). Thus, \(\mathbf{v}\) describes a fluid that is rotating (like a solid) about the \(z\)-axis with constant angular velocity \(\omega\). Show that \(\operatorname{div} \mathbf{v}=0\) and curl \(\mathbf{v}=2 \omega \mathbf{k}\).

Step-by-Step Solution

Answer

\(\operatorname{div} \mathbf{v} = 0\) and \(\nabla \times \mathbf{v} = 2\omega \mathbf{k}\).

Step 1: Understand Divergence

Divergence of a vector field \(\mathbf{v} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) in three dimensions is given by \(\operatorname{div} \mathbf{v} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\). In our case, \(P = -\omega y\), \(Q = \omega x\), and \(R = 0\). We must calculate each partial derivative.

Step 2: Compute Partial Derivatives for Divergence

Calculate \(\frac{\partial P}{\partial x} = \frac{\partial (-\omega y)}{\partial x} = 0\), \(\frac{\partial Q}{\partial y} = \frac{\partial (\omega x)}{\partial y} = 0\), and \(\frac{\partial R}{\partial z} = \frac{\partial (0)}{\partial z} = 0\).

Step 3: Sum Partial Derivatives for Divergence

Sum the partial derivatives to find \(\operatorname{div} \mathbf{v} = 0 + 0 + 0 = 0\). Thus, \(\operatorname{div} \mathbf{v} = 0\).

Step 4: Understand Curl

The curl of a vector field \(\mathbf{v} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is given by \(abla \times \mathbf{v} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k}\).

Step 5: Compute Partial Derivatives for Curl

Find \(\frac{\partial R}{\partial y} = 0\), \(\frac{\partial Q}{\partial z} = \, 0\), \(\frac{\partial P}{\partial z} = 0\), \(\frac{\partial R}{\partial x} = 0\), \(\frac{\partial Q}{\partial x} = \omega\), and \(\frac{\partial P}{\partial y} = -\omega\).

Step 6: Calculate Curl

Substitute these into the curl formula: \(abla \times \mathbf{v} = (0 - 0) \mathbf{i} + (0 - 0) \mathbf{j} + (\omega - (-\omega)) \mathbf{k} = 2\omega \mathbf{k}\).

Step 7: Conclusion of Curl

Thus, the curl of the vector field is \(abla \times \mathbf{v} = 2\omega \mathbf{k}\), confirming our solution.

Key Concepts

Divergence

Divergence in vector calculus is an operation that measures the magnitude of a source or sink at a given point in a vector field. You can think of it as the rate at which "stuff" is expanding out of a point. For a velocity field,

• positive divergence indicates a source, where the fluid is flowing outward,
• negative divergence indicates a sink, where the fluid is flowing inward,
• zero divergence indicates a constant-flow situation or a very balanced field.

The formula for divergence is \[ \operatorname{div} \mathbf{v} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]For the specific velocity field given by \( \mathbf{v}(x, y, z) = -\omega y \mathbf{i} + \omega x \mathbf{j} \), we found that each partial derivative of its components is zero, leading to a total divergence of zero. This tells us that the fluid is neither accumulating nor depleting in the region, indicating no net volume increase or decrease over time.

Curl

Curl is a vector operation that describes the rotation of a field. Imagine placing a tiny paddle wheel in the flow; curl measures whether it would spin, and in which direction. If you have a vector field representing a fluid,

• a non-zero curl would mean the fluid is rotating,
• a zero curl means the fluid is not rotating.

For our velocity field \( \mathbf{v} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the curl is defined as \[ abla \times \mathbf{v} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]By substituting in the components of the field, we determined that the curl results in \( 2 \omega \mathbf{k} \), indicating a rotational movement about the z-axis. The non-zero value suggests that the fluid possesses angular momentum, simulating the motion of spinning water.

Fluid Mechanics

Fluid mechanics is the field of study focused on fluids (liquids and gases) and how forces affect them. In this topic, understanding concepts like divergence and curl helps explain behaviors within fluid flows, such as rotation, expansion, or compression. By analyzing the velocity field of a fluid, engineers and scientists can predict light and detailed phenomena.
The velocity field given in the exercise is particularly typical in fluid dynamics. It is often used to illustrate rotational movement in a fluid poise where the fluid is moving in a circular path around an axis while not expanding or compressing in the total volume over time. Understanding these properties helps design turbines, predict weather patterns, and even understand ocean currents.

Velocity Field

A velocity field represents the speed and direction at each point in a fluid. Each point in the field expresses the local flow rate and direction that a fluid particle would take if it were at this point. Think of it as a map for particles; the vectors describe precisely where each particle goes and how fast.
In the given example, the velocity field \( \mathbf{v}(x, y, z) = -\omega y \mathbf{i} + \omega x \mathbf{j} \) describes a simple rotational flow. The terms \( -\omega y \) and \( \omega x \) control the circular motion around the z-axis, indicating how each component contributes to the direction.

• The \( \mathbf{i} \) term represents movement along the x-axis,
• the \( \mathbf{j} \) term for the y-axis.

Such models are essential for predicting how fluids influence their environments, such as determining paths in airflow around wings or water around ship hulls.