Problem 12 - Advanced Engineering Mathematics

Question

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=5 y^{3} \mathbf{i}+\left(\frac{1}{2} x^{3} y^{2}-x y\right) \mathbf{j}-\left(x^{3} y z-x z\right) \mathbf{k} $$

Step-by-Step Solution

Answer

The curl is \((-x^3z, 3x^2yz - z, \frac{3}{2}x^2y^2 - 16y^2)\) and the divergence is \(0\).

Step 1: Identify the components of the vector field

The given vector field is \( \mathbf{F}(x, y, z) = 5y^3 \mathbf{i} + \left(\frac{1}{2}x^3y^2 - xy\right) \mathbf{j} + \left(-x^3yz + xz\right) \mathbf{k} \). Here, \( F_1 = 5y^3 \), \( F_2 = \frac{1}{2}x^3y^2 - xy \), and \( F_3 = -x^3yz + xz \).

Step 2: Formula for Curl

The curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the determinant:\[\mathrm{curl} \, \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}\]Substitute \( P = 5y^3 \), \( Q = \frac{1}{2}x^3y^2 - xy \), \( R = -x^3yz + xz \) into the determinant.

Step 3: Calculate Partial Derivatives for Curl

Calculate the partial derivatives:- \( \frac{\partial R}{\partial y} = -x^3z \)- \( \frac{\partial Q}{\partial z} = 0 \)- \( \frac{\partial P}{\partial z} = 0 \)- \( \frac{\partial R}{\partial x} = -3x^2yz + z \)- \( \frac{\partial Q}{\partial x} = \frac{3}{2}x^2y^2 - y \)- \( \frac{\partial P}{\partial y} = 15y^2 \)

Step 4: Solve for Each Component of the Curl

Substitute the partial derivatives into the formula to find:- The \( \mathbf{i} \) component: \( \frac{\partial}{\partial y}R - \frac{\partial}{\partial z}Q = -x^3z - 0 = -x^3z \)- The \( \mathbf{j} \) component: \( \frac{\partial}{\partial z}P - \frac{\partial}{\partial x}R = 0 - (-3x^2yz + z) = 3x^2yz - z \)- The \( \mathbf{k} \) component: \( \frac{\partial}{\partial x}Q - \frac{\partial}{\partial y}P = \left(\frac{3}{2}x^2y^2 - y\right) - 15y^2 = \frac{3}{2}x^2y^2 - 16y^2 \)Therefore, the curl is \( \mathrm{curl} \, \mathbf{F} = (-x^3z)\mathbf{i} + (3x^2yz - z)\mathbf{j} + \left(\frac{3}{2}x^2y^2 - 16y^2\right)\mathbf{k} \).

Step 5: Formula for Divergence

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by:\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]

Step 6: Calculate Partial Derivatives for Divergence

Using the components of \( \mathbf{F} \):- \( \frac{\partial (5y^3)}{\partial x} = 0 \)- \( \frac{\partial}{\partial y}\left(\frac{1}{2}x^3y^2 - xy\right) = x^3y - x \)- \( \frac{\partial}{\partial z}(-x^3yz + xz) = -x^3y + x \)

Step 7: Solve for Divergence

Sum the partial derivatives to find the divergence:\( abla \cdot \mathbf{F} = 0 + (x^3y - x) + (-x^3y + x) = 0 \).Thus, the divergence of \( \mathbf{F} \) is zero.

Key Concepts

Curl of a Vector Field

Curl is a concept in vector calculus and measures the tendency of flow to rotate around a point in a vector field. When we talk about the curl of a vector field, we are examining how the field twists or rotates. This is particularly important in fields like fluid dynamics, where we want to know if the flow around a point has a swirling pattern.
To calculate the curl, we use a determinant involving partial derivatives. The formula for the curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by:

• \( \mathrm{curl} \, \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \)

In our example, to find the curl, we compute the determinant by substituting the components of \( \mathbf{F} \) into the formula and calculating partial derivatives for \( P, Q, \) and \( R \).
Understanding curl helps in fields like electromagnetism and fluid mechanics by indicating regions where rotational forces or movements may be present.

Divergence of a Vector Field

Divergence is another key concept in vector calculus. It measures a vector field's tendency to originate from or converge into a point. Imagine a vector field as wind blowing through a region of space; divergence tells us how much air is spreading out from a point.
The formula for the divergence \( abla \cdot \mathbf{F} \) of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is:

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

This sum of partial derivatives gives us a scalar quantity representing either a source or a sink type behavior at a given point in the field.
In our exercise, after calculating the necessary partial derivatives, we sum them to obtain the divergence. It's found to be zero, which often indicates that the vector field is solenoidal, meaning there are no sources or sinks within the region.

Partial Derivatives

Partial derivatives are fundamental in vector calculus as they measure how a function changes as individual variables change, while others are held constant. This is crucial in understanding how multivariable functions behave.
In our context, each vector field component requires partial derivatives with respect to the variables \( x, y, \) and \( z \) to compute both the curl and divergence. Here's how we approach partial derivatives:

• Determine how \( P, Q, \) and \( R \) change with each variable individually.
• Calculate these derivatives separately.
• Use these derived values in further calculations, such as computing the curl or divergence.

In the original problem, partial derivatives are used to quantify these changes, allowing us to plug them into the formulas for curl and divergence. Mastering partial derivatives is essential for any deeper exploration of vector calculus applications.