Problem 9 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) is \( \mathbf{0} \) and its divergence is \( 4y + 8z \).

Step 1: Identify the Vector Field Components

The given vector field is \( \mathbf{F}(x, y, z) = 4xy \mathbf{i} + (2x^2 + 2yz) \mathbf{j} + (3z^2 + y^2) \mathbf{k} \). Identify the components as follows: \( F_1 = 4xy \), \( F_2 = 2x^2 + 2yz \), and \( F_3 = 3z^2 + y^2 \).

Step 2: Compute the Curl of \( \mathbf{F} \)

The curl of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).Calculate each component:- \( \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(3z^2 + y^2) = 2y \)- \( \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(2x^2 + 2yz) = 2y \)- \( \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(3z^2 + y^2) = 0 \)- \( \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(4xy) = 0 \)- \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(2x^2 + 2yz) = 4x \)- \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(4xy) = 4x \)Thus, \( abla \times \mathbf{F} = (2y - 2y)\mathbf{i} - (0 - 0)\mathbf{j} + (4x - 4x)\mathbf{k} = 0 \mathbf{i} - 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{0} \).

Step 3: Compute the Divergence of \( \mathbf{F} \)

The divergence of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).Calculate each term:- \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(4xy) = 4y \)- \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(2x^2 + 2yz) = 2z \)- \( \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(3z^2 + y^2) = 6z \)Thus, \( abla \cdot \mathbf{F} = 4y + 2z + 6z = 4y + 8z \).

Key Concepts

Curl of a Vector Field

The curl of a vector field helps us understand the rotation or "twisting" of the field at a point. It is a measure of how the vector field rotates around any given point.
To find the curl of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), we use the formula:

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

When we calculate these partial derivatives, we find out if there is any rotational effect. In our original exercise, the components end up being zero, which implies that the vector field has no rotation at any point. This is quite significant in fields like fluid dynamics, where it can indicate whether a fluid is circulating around an axis or not.

Divergence of a Vector Field

The divergence of a vector field is an important concept as it indicates the rate at which "stuff" is expanding out of a point or flowing into it. It essentially tells us how the vector field spreads out from or converges into a point.
To compute the divergence \( abla \cdot \mathbf{F} \) for a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), you need to sum up the partial derivatives of each component:

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

For the vector field in our example, calculating these derivations gave us \( 4y + 8z \). This result indicates how the vector field expands or contracts at any point \((x, y, z)\), providing insight into whether sources or sinks are present in the field.

Vector Field Components

Understanding the components of a vector field is crucial before diving into operations like curl and divergence. These components are essentially the building blocks, representing the vector field's effect in different spatial directions.
For a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), each component \( P \), \( Q \), and \( R \) represents the vector's influence in the \( x \), \( y \), and \( z \) directions respectively:

• \( F_1(x, y, z) = 4xy \): The component along the \( x \)-direction.
• \( F_2(x, y, z) = 2x^2 + 2yz \): The component along the \( y \)-direction.
• \( F_3(x, y, z) = 3z^2 + y^2 \): The component along the \( z \)-direction.

Each of these functions depends on \( x \), \( y \), or \( z \), pointing to how the vector field changes with those variables. This understanding sets the stage for more complex operations like finding the curl or divergence, highlighting the dynamic nature of the vector field.