Problem 15 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The curl is \( \nabla \times \mathbf{F} = \left( x(2y e^y + y^2 e^y) + x^3 y (e^z + ze^z) \right) \mathbf{i} - y^2 e^y \mathbf{j} + (-3x^2 y z e^z - x e^x) \mathbf{k} \), and the divergence is \( y e^x (1 + x) - x^3 z e^z \).

Step 1: Understanding the Vector Field

The vector field given is \( \mathbf{F}(x, y, z) = x y e^{x} \mathbf{i} - x^{3} y z e^{z} \mathbf{j} + x y^{2} e^{y} \mathbf{k} \). It is composed of three components: \( F_1 = x y e^{x} \), \( F_2 = -x^{3} y z e^{z} \), and \( F_3 = x y^{2} e^{y} \). We will use these components to find the curl and the divergence.

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

The curl of a vector field \( \mathbf{F} = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \) is given by:\[ abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \]Calculating each component:- \( \frac{\partial F_3}{\partial y} = \frac{\partial}{\partial y} (x y^2 e^y) = x (2y e^y + y^2 e^y) \)- \( \frac{\partial F_2}{\partial z} = \frac{\partial}{\partial z} (-x^3 y z e^z) = -x^3 y (e^z + ze^z) \)Therefore, the \( \mathbf{i} \)-component is:\[ x(2y e^y + y^2 e^y) + x^3 y (e^z + ze^z) \]- \( \frac{\partial F_1}{\partial z} = 0 \)- \( \frac{\partial F_3}{\partial x} = y^2 e^y \)Therefore, the \( \mathbf{j} \)-component is:\[ 0 - y^2 e^y = -y^2 e^y \]- \( \frac{\partial F_2}{\partial x} = -3x^2 y z e^z \)- \( \frac{\partial F_1}{\partial y} = x e^x \)Therefore, the \( \mathbf{k} \)-component is:\[ -3x^2 y z e^z - x e^x \]Thus, the curl is:\[ abla \times \mathbf{F} = \left( x(2y e^y + y^2 e^y) + x^3 y (e^z + ze^z) \right) \mathbf{i} - y^2 e^y \mathbf{j} + (-3x^2 y z e^z - x e^x) \mathbf{k} \]

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

The divergence of a vector field \( \mathbf{F} = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \) is given by:\[ abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \]Calculating each component:- \( \frac{\partial F_1}{\partial x} = y e^x + xy e^x = y e^x (1 + x) \)- \( \frac{\partial F_2}{\partial y} = -x^3 z e^z \)- \( \frac{\partial F_3}{\partial z} = 0 \)Thus, the divergence is:\[ abla \cdot \mathbf{F} = y e^x (1 + x) - x^3 z e^z \]

Key Concepts

Curl of a Vector Field

The curl of a vector field is a concept from vector calculus that indicates the rotational tendency of the field. If you imagine the vector field as the flow of a fluid, the curl tells you how much and in what direction the fluid is rotating around a point.

To calculate the curl of a vector field \( \mathbf{F}(x, y, z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \), we use the formula:

• \( \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} \)
• \( \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} \)
• \( \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \)

These components give us the vector of the curl. Each component of this vector represents the rotation in a specific plane:

• The \( \mathbf{i} \)-component reflects rotation in the \( yz \)-plane.
• The \( \mathbf{j} \)-component reflects rotation in the \( zx \)-plane.
• The \( \mathbf{k} \)-component reflects rotation in the \( xy \)-plane.

Finding the curl involves computing partial derivatives of these components, as detailed in the solution steps. The final curl vector provides a comprehensive look at the field's rotational aspects.

Divergence of a Vector Field

Divergence is another fundamental concept in vector calculus. It measures how much a vector field spreads out or diverges from a point.

If the vector field is thought of as representing flow, divergence at a certain point indicates whether the flow is expanding or compressing at that point.

The mathematical formula for the divergence of a vector field \( \mathbf{F}(x, y, z) = F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k} \) is given by the sum of the partial derivatives:

• \( \frac{\partial F_1}{\partial x} \)
• \( \frac{\partial F_2}{\partial y} \)
• \( \frac{\partial F_3}{\partial z} \)

By adding these derivatives together, you can determine how much the vector field diverges or converges around a specific point. A positive divergence indicates a source (flow spreading out) while a negative divergence indicates a sink (flow converging in).

The original exercise showed how to compute each of these partial derivatives, adding them to find the total divergence of the field. It's a snapshot of overall expansion or convergence at any desired location in the field.

Partial Derivatives

Partial derivatives are a key concept when working with functions involving multiple variables, such as those found in vector fields.

In simple terms, a partial derivative measures how a function changes as one variable changes, while keeping the other variables constant. This is particularly useful in vector calculus because vector fields often depend on more than one variable.

To compute a partial derivative, you fix all other variables and differentiate with respect to the variable of interest:

• To find \( \frac{\partial f}{\partial x} \), treat all other variables \( y, z, \ldots \) as constants and differentiate with respect to \( x \).
• Similarly, for \( \frac{\partial f}{\partial y} \), hold other variables constant and differentiate with respect to \( y \).

In solving the original exercise, partial derivatives were taken of the components of the vector field to calculate the curl and divergence. Each step involved isolating a variable and finding how the function changes in that direction.

This approach helps break down complex problems into simpler parts, making it possible to analyze multi-variable functions efficiently.