Problem 7 - Advanced Engineering Mathematics

Question

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x z \mathbf{i}+y z \mathbf{j}+x y \mathbf{k} $$

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) is \( (x-y) \mathbf{i} + (x-y) \mathbf{j} \), and the divergence is \( 2z \).

Step 1: Understanding the Definitions

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

Step 2: Identifying Components of the Vector Field

For the vector field \( \mathbf{F}(x, y, z) = x z \mathbf{i} + y z \mathbf{j} + x y \mathbf{k} \), identify \( P = xz \), \( Q = yz \), and \( R = xy \).

Step 3: Calculating the Curl

Calculate each component of the curl using the identified \( P \), \( Q \), and \( R \) from Step 2: 1. \( \frac{\partial R}{\partial y} = x \) and \( \frac{\partial Q}{\partial z} = y \) so the \( \mathbf{i} \)-component is \( x - y \). 2. \( \frac{\partial P}{\partial z} = x \) and \( \frac{\partial R}{\partial x} = y \) so the \( \mathbf{j} \)-component is \( x - y \). 3. \( \frac{\partial Q}{\partial x} = 0 \) and \( \frac{\partial P}{\partial y} = 0 \) so the \( \mathbf{k} \)-component is \( 0 \). Thus, \( abla \times \mathbf{F} = (x-y) \mathbf{i} + (x-y) \mathbf{j} \).

Step 4: Calculating the Divergence

Calculate the divergence using the components found in Step 2: 1. \( \frac{\partial P}{\partial x} = z \) 2. \( \frac{\partial Q}{\partial y} = z \) 3. \( \frac{\partial R}{\partial z} = 0 \) Sum these to find \( abla \cdot \mathbf{F} = z + z + 0 = 2z \).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a way to measure the rotation at a certain point within that field. Think of it like capturing how much the field "twirls" around that point. It's very much like the way water swirls around when you stir it.

The mathematical tool we use for this idea is the curl vector operator, which is denoted by \( abla \times \mathbf{F} \). In simpler terms, you can think of it as a special cross product that involves partial derivatives of the field components. For a given vector field defined by three functions \( P, Q, \) and \( R \), the curl is calculated as:
\( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).

For example, if we have a vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \), the calculation goes as follows:

• For the \( \mathbf{i} \)-component: Calculate \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = x - y \).
• For the \( \mathbf{j} \)-component: Calculate \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = x - y \).
• For the \( \mathbf{k} \)-component: Calculate \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 \).

The result is the curl vector \( (x-y) \mathbf{i} + (x-y) \mathbf{j} \), which reflects how the original field "rotates" around points within it.

Divergence of a Vector Field

Divergence is a scalar measure that gives us an idea of how much a vector field spreads out from or converges to a point. Imagine it as the field's "net outflow" or "net inflow" at that point, similar to how water spreads out or converges.

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

Applying this to our vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \), we follow these steps:

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

When you sum these components, you get the final divergence value: \( abla \cdot \mathbf{F} = z + z + 0 = 2z \). This value, \( 2z \), indicates how much the vector field "diverges" from each point (or converges, if the value were negative) at any place within the field.

Vector Field Components

Understanding vector field components is crucial for computing both curl and divergence. A vector field in three-dimensional space can be visualized as an array of vectors, each associated with a unique set of coordinates \((x, y, z)\).

The vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \) is defined by its components along the x, y, and z directions. These components are:

• \( P(x, y, z) = xz \), the x-component.
• \( Q(x, y, z) = yz \), the y-component.
• \( R(x, y, z) = xy \), the z-component.

Recognizing and splitting a vector field into these components allows us to perform necessary operations like curl and divergence. Each component showcases how the vector field behaves in its respective direction.

This breakdown not only helps in analytical calculations but also lets us visualize the physical phenomenon being modeled, whether it's electromagnetic fields, fluid flows, or force fields. Being able to identify and work with vector components is foundational in learning vector calculus, as it serves as the building block for more complex operations and analyses.