Problem 372 - Calculus Volume 3

Question

Velocity field \(\quad \mathbf{v}=\left\langle 0,1-x^{2}, 0\right\rangle, \quad\) for \(|x| \leq 1\) and \(|z| \leq 1,\) represents a horizontal flow in the \(y\) -direction. Compute the curl of \(\mathbf{v}\) in a clockwise rotation.

Step-by-Step Solution

Answer

The curl of \(\mathbf{v}\) is \(\left\langle 0, 0, -2x \right\rangle\).

Step 1: Understanding Curl Definition

The curl of a vector field \( \mathbf{F} = \left\langle P, Q, R \right\rangle \) is given by the formula \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \). We need to identify \(P, Q,\) and \(R\) from the vector \( \mathbf{v} = \left\langle 0, 1-x^{2}, 0 \right\rangle \).

Step 2: Identifying Components of \(\mathbf{v}\)

For the given vector field \( \mathbf{v} = \left\langle 0, 1-x^{2}, 0 \right\rangle \), we identify \( P = 0 \), \( Q = 1-x^{2} \), and \( R = 0 \).

Step 3: Calculating Partial Derivatives

We calculate the required partial derivatives: \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(1-x^2) = -2x \), \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(0) = 0 \), \( \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(0) = 0 \), \( \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(1-x^2) = 0 \), \( \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(0) = 0 \), \( \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(0) = 0 \).

Step 4: Applying Curl Formula

Substitute the partial derivatives into the curl formula: \( abla \times \mathbf{v} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) = \left( 0 - 0, 0 - 0, -2x - 0 \right) = \left\langle 0, 0, -2x \right\rangle \).

Step 5: Interpreting the Result

The resulting vector \( \left\langle 0, 0, -2x \right\rangle \) indicates the direction and magnitude of the curl. A negative \( z \)-component signifies a clockwise rotation in the \( xy \)-plane. Therefore, the curl of \( \mathbf{v} \) is consistent with a clockwise rotation around the \( z \)-axis.

Key Concepts

Curl of a Vector Field

In vector calculus, recognizing the curl of a vector field is an important task when exploring the rotation of fluid flows or electromagnetic fields. The curl is essentially a measure of the twisting or angular momentum at a point in the field. For the vector field \( \mathbf{v} = \left\langle 0, 1-x^2, 0 \right\rangle \), it represents a flow moving primarily in the \( y \)-direction. The formula for the curl \( abla \times \mathbf{F} \) is:
\[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\]This tool helps to identify the local spinning effect of the vector field around a point. By understanding how each components \( P \), \( Q \), and \( R \) contribute to this rotation, we can determine the twisting behavior of the vector field.
This can be especially useful in applications such as fluid dynamics to determine the rotational forces present at various points.

Partial Derivatives

Partial derivatives serve as the backbone in calculating changes in multi-dimensional functions, which include vector fields. These derivatives give us the rate of change of a function with respect to one variable while keeping the other variables constant. For our velocity field \( \mathbf{v} = \left\langle 0, 1-x^2, 0 \right\rangle \), it's important to identify:

• \( \frac{\partial Q}{\partial x} = -2x \)
• \( \frac{\partial P}{\partial y} = 0 \)
• \( \frac{\partial R}{\partial y} = 0 \)
• \( \frac{\partial Q}{\partial z} = 0 \)
• \( \frac{\partial P}{\partial z} = 0 \)
• \( \frac{\partial R}{\partial x} = 0 \)

These calculations allow us to understand how the vector field behaves in different directions, helping to pinpoint how changes in various axes affect the field's properties.
In general, partial derivatives are crucial in a wide array of calculus applications from physics to engineering, providing foundational insight into how functions behave in a multi-variable environment.

Velocity Field Analysis

Velocity field analysis is all about examining how objects move through a space, which is crucial in fields like fluid dynamics, meteorology, and aerodynamics. In this exercise, analyzing the velocity field \( \mathbf{v} = \left\langle 0, 1-x^2, 0 \right\rangle \) simplifies to assessing the behavior of a horizontal flow in the \( y \)-direction.
The velocity field illustrates that all flow happens along \( y \) and is dependent only on \( x \).
Specifically:

• The flow is strongest where \( x = 0 \), and dies out as \( |x| \) approaches 1.
• The value inside the vector signifies that flow rate diminishes with increasing \( x \) magnitude.
• The absence of \( z \)-component means no flow or change happens in the \( z \)-direction.

Velocity fields tell us about not only where matter is going but provide insight into how forces might act in engineering or environmental scenarios. They offer a map showing speed and direction that is decisive for problem-solving in real-world applications. This examination helps build a clearer picture of the occurring dynamics and facilitates better design and prediction outcomes.