Problem 10 - Advanced Engineering Mathematics

Question

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=(x-y)^{3} \mathbf{i}+e^{-y z} \mathbf{j}+x y e^{2 y} \mathbf{k} $$

Step-by-Step Solution

Answer

Curl: \( \nabla \times \mathbf{F} = (xe^{2y} + 2xye^{2y} + ye^{-yz}) \mathbf{i} - (ye^{2y}) \mathbf{j} + 3(x-y)^2 \mathbf{k} \); Divergence: \( \nabla \cdot \mathbf{F} = 3(x-y)^2 - ze^{-yz} \).

Step 1: Understand the Vector Field and Notations

The vector field is given as \( \mathbf{F}(x, y, z)=(x-y)^3 \mathbf{i} + e^{-yz} \mathbf{j} + xy e^{2y} \mathbf{k} \). The task is to find both the curl \( abla \times \mathbf{F} \) and the divergence \( abla \cdot \mathbf{F} \) of this vector field.

Step 2: Calculate the Curl

Use the formula \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} - \left( \frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \). For \( \mathbf{F} \), we identify \( F_1 = (x-y)^3 \), \( F_2 = e^{-yz} \), \( F_3 = xye^{2y} \).Calculate each component:- \( \frac{\partial F_3}{\partial y} = x e^{2y} + 2xy e^{2y} \)- \( \frac{\partial F_2}{\partial z} = -ye^{-yz} \)- \( \frac{\partial F_3}{\partial x} = ye^{2y} \)- \( \frac{\partial F_1}{\partial z} = 0 \)- \( \frac{\partial F_2}{\partial x} = 0 \)- \( \frac{\partial F_1}{\partial y} = -3(x-y)^2 \)Substitute these into the curl formula:\[ abla \times \mathbf{F} = \left( x e^{2y} + 2xy e^{2y} + y e^{-yz} \right) \mathbf{i} - \left( ye^{2y} \right) \mathbf{j} + \left( 3(x-y)^2 \right) \mathbf{k} \]

Step 3: Calculate the Divergence

Use the formula \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). Calculate each component:- \( \frac{\partial F_1}{\partial x} = 3(x-y)^2 \)- \( \frac{\partial F_2}{\partial y} = -ze^{-yz} \)- \( \frac{\partial F_3}{\partial z} = 0 \)Substitute these into the divergence formula:\[ abla \cdot \mathbf{F} = 3(x-y)^2 - ze^{-yz} \]

Step 4: Verify the Solution

Review each differentiation step and computation for accuracy. Make sure the partial derivatives applied match the specified components. This ensures that both curl and divergence calculations align with the vector field's definition.

Key Concepts

Curl

The curl of a vector field is a measure of the rotation or "twirl" of vectors in the field. Imagine swirling water; the faster or tighter the water swirls at a point, the more curl there is. Mathematically, the curl is a vector and is calculated using the cross product of the nabla operator (∇) and the vector field. For a vector field \( \mathbf{F}(x, y, z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \), the curl is given by:

• \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} - \left( \frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \).

In simple terms, each component acts like a mini derivative that measures changes of other components, capturing rotational tendencies. Exploring real-world contexts, curl is crucial in fluid dynamics and electromagnetic theory, where understanding rotation within fields helps manage energy flow or design machinery.

Divergence

Divergence offers insight into the "outflow" or "inflow" at a point in a vector field and is a tool for assessing the source or sink nature of a point. Imagine air radiating from a heater; divergence indicates whether air (or a field) spreads out or converges at a particular point.
The mathematical representation of divergence in a vector field \( \mathbf{F} \) is:

• \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \).

Using divergence, we can predict the net flow or provide a measure of expansion within a medium, which is fundamental in fluid mechanics, thermodynamics, and electromagnetism. For instance, in an air pressure field, areas of positive divergence could indicate sources, perhaps where air is being introduced.

Vector Fields

Vector fields attach a vector to every point in space, resembling a force pushing or a velocity moving across each point. They are common in physics and engineering, representing forces like gravity, wind, or electromagnetic fields.
A vector field can be written compactly as \( \mathbf{F}(x, y, z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \). Each component \( F_1, F_2, F_3 \) depends on spatial coordinates, reflecting the vector’s direction and magnitude at each position.

• \( F_1 \): the x-component
• \( F_2 \): the y-component
• \( F_3 \): the z-component

In practice, vector fields help model and solve problems involving fluid flow, electromagnetic forces, and gravitational effects, among others. Understanding the basics like curl and divergence enhances your ability to analyze these fields and predict physical phenomena.