Problem 12 - Advanced Engineering Mathematics

Question

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

Curl: \(-x^3z\mathbf{i} + (3x^2yz - z)\mathbf{j} + \left(\frac{3}{2}x^2y^2 - 16y^2\right)\mathbf{k}\); Divergence: \(-x\).

Step 1: Reminder of Curl Formula

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

Step 2: Compute Partial Derivatives for Curl

For \( \mathbf{F}(x, y, z) \):- \( P = 5y^3 \), \( Q = \frac{1}{2}x^3y^2 - xy \), \( R = -x^3yz + xz \).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 3: Calculate Curl Components

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

Step 4: Calculate the Curl

Combine the components from Step 3:\[ abla \times \mathbf{F} = -x^3z \mathbf{i} + (3x^2yz - z) \mathbf{j} + \left(\frac{3}{2}x^2y^2 - 16y^2\right) \mathbf{k}. \]

Step 5: Reminder of Divergence Formula

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: Compute Partial Derivatives for Divergence

Partial derivatives:- \( \frac{\partial P}{\partial x} = 0 \) as \( P \) is independent of \( x \).- \( \frac{\partial Q}{\partial y} = x^3y - x \).- \( \frac{\partial R}{\partial z} = -x^3y \).

Step 7: Calculate the Divergence

Substitute the partial derivatives into the divergence formula:\[ abla \cdot \mathbf{F} = 0 + x^3y - x - x^3y = -x. \]

Step 8: Final Result

Thus, the curl of the vector field is \[ abla \times \mathbf{F} = -x^3z \mathbf{i} + (3x^2yz - z) \mathbf{j} + \left(\frac{3}{2}x^2y^2 - 16y^2\right) \mathbf{k}, \]and the divergence is \[ abla \cdot \mathbf{F} = -x. \]

Key Concepts

Curl of a Vector Field

Curl is a fundamental concept in vector calculus used to describe the rotation of a vector field. When you find the curl of a vector field, you essentially measure how much the field is "twisting" around a point. In physics, it can represent the circulation density or rotational behavior of the vector field.

To understand curl, consider the vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \). The formula for calculating the curl is given by:

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

To calculate the curl of \( \mathbf{F}(x, y, z) = 5y^3 \mathbf{i} + \left(\frac{1}{2} x^3 y^2 - xy\right) \mathbf{j} - \left(x^3 yz - xz\right) \mathbf{k} \), follow these steps:

• Calculate the necessary partial derivatives of \( R \) with respect to \( y \), \( Q \) with respect to \( z \), etc.
• Substitute all partial derivatives into the curl formula.
• Combine results to get the vector for the curl.

Curl gives you a new vector, demonstrating how each small piece of the original field rotates around an axis.

Divergence of a Vector Field

Divergence helps us understand how a vector field behaves in terms of expansion or contraction. It can be thought of as a measure of how much a field is spreading out from or converging into a point.

To find the divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), use the divergence formula:

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

Let's consider the same vector field \( \mathbf{F}(x, y, z) = 5y^3 \mathbf{i} + \left(\frac{1}{2} x^3 y^2 - xy\right) \mathbf{j} - \left(x^3 yz - xz\right) \mathbf{k} \):

• Calculate partial derivatives: \( \frac{\partial P}{\partial x}, \frac{\partial Q}{\partial y}, \) and \( \frac{\partial R}{\partial z} \).
• Add these derivatives together according to the divergence formula.

The result is a scalar. In the example, the divergence \( abla \cdot \mathbf{F} \) resulted in \(-x\), demonstrating how the field decreases or "shrinks" around a given point by \(-x\).

Partial Derivatives

Partial derivatives are crucial in vector calculus and are used when examining how a function changes as you vary one of its variables, while keeping other variables constant. This concept is vital for understanding curl and divergence since these operations rely heavily on partial derivatives.

Calculating a partial derivative is similar to finding an ordinary derivative, but you treat all other variables as constants:

• For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \).
• Keep \( y \) and \( z \) constant, while differentiating the part of the function dependent on \( x \).

In the problem, several partial derivatives were calculated:

• \( \frac{\partial P}{\partial y} = 15y^2 \)
• \( \frac{\partial Q}{\partial x} = \frac{3}{2}x^2y^2 - y \)
• \( \frac{\partial R}{\partial z} = -x^3y \)

These derivatives were used to solve for the curl and divergence. Mastering partial derivatives involves practice, but they provide powerful insights into the behavior of multivariable functions.