Problem 7 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Vector Field

We have the vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \). The components are \( F_x = xz \), \( F_y = yz \), and \( F_z = xy \). We need to calculate both the curl and the divergence of this vector field.

Step 2: Formula for Curl and Divergence

The curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \) is defined as \( abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \). The divergence is \( abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \).

Step 3: Calculating Divergence

For divergence: \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(xz) + \frac{\partial}{\partial y}(yz) + \frac{\partial}{\partial z}(xy) \). This gives \( z + z + 0 = 2z \). Thus, the divergence of \( \mathbf{F} \) is \( 2z \).

Step 4: Calculating Curl

For curl: Compute each component separately. 1) \( \frac{\partial F_z}{\partial y} = x \) and \( \frac{\partial F_y}{\partial z} = y \), so the \( \mathbf{i} \) component is \( x - y \). 2) \( \frac{\partial F_x}{\partial z} = x \) and \( \frac{\partial F_z}{\partial x} = y \), so the \( \mathbf{j} \) component is \( x - y \). 3) \( \frac{\partial F_y}{\partial x} = 0 \) and \( \frac{\partial F_x}{\partial y} = 0 \), so the \( \mathbf{k} \) component is \( 0 - 0 = 0 \). Therefore, \( abla \times \mathbf{F} = (x-y) \mathbf{i} + (x-y) \mathbf{j} + 0 \mathbf{k} \).

Step 5: Summary of Results

We found that the divergence of the vector field \( \mathbf{F} \) is \( 2z \) and the curl is \((x-y) \mathbf{i} + (x-y) \mathbf{j} \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations on them. It's essential for fields such as physics and engineering that study quantities with both magnitude and direction.
Two crucial operations used in vector calculus are **curl** and **divergence**. Curl measures the rotation of a vector field, while divergence measures the rate at which "stuff" is expanding or contracting at a point.
To compute the curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), we use the formula:

• \(abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k}\).

For divergence, we use:

• \(abla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\).

These operations help in understanding the dynamics of vector fields by providing insight into their behavior and properties.

Vector Fields

A vector field is a function that assigns a vector to every point in space. These vectors could represent things like velocity in fluids or force in electromagnetics.
For example, a vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \) assigns a vector to each point \((x, y, z)\) in space.
Key properties of vector fields include:

• **Magnitude**: Refers to the length of the vectors generated by the field at different points.
• **Direction**: Indicates where the vector is pointing at each point.
Understanding these aspects helps in visualizing how the vector field behaves and influences the space it occupies.

Partial Derivatives

Partial derivatives are used to investigate how a function changes as each variable is varied independently, holding others constant.
In a multi-variable function like a vector field \( \mathbf{F}(x, y, z) = xz \mathbf{i} + yz \mathbf{j} + xy \mathbf{k} \), partial derivatives are crucial.

• The partial derivative \( \frac{\partial}{\partial x} \) looks at the change concerning \( x \), treating \( y \) and \( z \) as constants.
• Similarly, \( \frac{\partial}{\partial y} \) and \( \frac{\partial}{\partial z} \) evaluate changes with respect to \( y \) and \( z \) respectively.

These derivatives provide the foundation for calculating curl and divergence, revealing how vector fields fluctuate and rotate at any given point.