Problem 13 - Advanced Engineering Mathematics

Question

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x e^{-z} \mathbf{i}+4 y z^{2} \mathbf{j}+3 y e^{-z} \mathbf{k} $$

Step-by-Step Solution

Answer

Curl: \( (3 e^{-z} - 8 y z) \mathbf{i} - x e^{-z} \mathbf{j} \). Divergence: \( e^{-z} + 4 z^2 - 3 y e^{-z} \).

Step 1: Review the Formulas

To find the curl and divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), recall the formulas: The curl is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \). The divergence is \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).

Step 2: Identify Components P, Q, R

Identify the components of the vector field \( \mathbf{F}(x, y, z) = x e^{-z} \mathbf{i} + 4 y z^2 \mathbf{j} + 3 y e^{-z} \mathbf{k} \), where \( P = x e^{-z} \), \( Q = 4 y z^2 \), and \( R = 3 y e^{-z} \).

Step 3: Calculate the Curl Components

Calculate each component of the curl using the identified P, Q, R.1. \( \frac{\partial R}{\partial y} = 3 e^{-z} \)2. \( \frac{\partial Q}{\partial z} = 8 y z \)3. \( \frac{\partial P}{\partial z} = -x e^{-z} \)4. \( \frac{\partial R}{\partial x} = 0 \)5. \( \frac{\partial Q}{\partial x} = 0 \)6. \( \frac{\partial P}{\partial y} = 0 \)Now compute the curl: \[ (3 e^{-z} - 8 y z) \mathbf{i} + (-x e^{-z} - 0) \mathbf{j} + (0 - 0) \mathbf{k} \].

Step 4: Simplify the Curl

Simplify the curl expression: \[ abla \times \mathbf{F} = (3 e^{-z} - 8 y z) \mathbf{i} - x e^{-z} \mathbf{j} \].

Step 5: Calculate the Divergence

Calculate the divergence using partial derivatives: \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).- \( \frac{\partial P}{\partial x} = e^{-z} \)- \( \frac{\partial Q}{\partial y} = 4 z^2 \)- \( \frac{\partial R}{\partial z} = -3 y e^{-z} \)So, \( abla \cdot \mathbf{F} = e^{-z} + 4 z^2 - 3 y e^{-z} \).

Step 6: Summarize the Results

The final expressions for the curl and divergence are:Curl: \( abla \times \mathbf{F} = (3 e^{-z} - 8 y z) \mathbf{i} - x e^{-z} \mathbf{j} \).Divergence: \( abla \cdot \mathbf{F} = e^{-z} + 4 z^2 - 3 y e^{-z} \).

Key Concepts

Curl of a Vector Field

In vector calculus, the curl of a vector field is a vital concept used to measure the rotation at a point in the field. Think of it as a measure of how much and in what direction a field circulates around a point. If you imagine a small paddle wheel placed in the vector field, the curl represents how quickly the wheel is spinning.

The formula to find the curl of a vector field \( \mathbf{F} \) with components \( P, Q, R \) is:

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

The components of the curl vector tell us the rotation around the respective axes:

• The \( \mathbf{i} \) component indicates rotation around the x-axis.
• The \( \mathbf{j} \) component around the y-axis.
• The \( \mathbf{k} \) component around the z-axis.

Understanding the curl helps especially in fluid dynamics and electromagnetism where field rotations are crucial aspects.

Divergence of a Vector Field

Divergence is another essential concept in vector calculus used to measure the magnitude of a field's source or sink at a given point. It's like checking how much a fluid is expanding or contracting at a point. Imagine a tiny balloon that inflates or deflates depending on the divergence of a fluid's velocity field surrounding it.

To calculate the divergence of the vector field \( \mathbf{F} \), use the formula:

• \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \)

Each partial derivative term represents how much the component of the vector field is expanding or contracting in the respective axis direction.

A positive divergence signifies a source where flow is spreading out, while a negative divergence indicates a sink where flow converges. Incompressible fluids, such as ideal fluids, exhibit zero divergence.

Partial Derivatives

Partial derivatives are the heart of multivariable calculus, providing a way to see how a function changes as one of its input variables changes while keeping the other variables constant. They allow you to focus on one direction or dimension at a time, which is beneficial in visualization and practical applications.

For a function of several variables, \( f(x, y, z) \), the partial derivative with respect to one of the variables, say \( x \), is noted as \( \frac{\partial f}{\partial x} \). It measures the rate of change of the function in the x-direction, holding \( y \) and \( z \) fixed.

Consider how you walk across a hilly landscape: the slope or steepness you feel underfoot is the partial derivative of the hill's height (elevation) with respect to your walking direction.

• It provides critical information for computing the curl and divergence as seen in the above vector calculus topics.
• Each partial derivative contributes to understanding various changes in different directions, essential in machine learning, physics, and engineering.