Problem 11 - Advanced Engineering Mathematics

Question

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=3 x^{2} y \mathbf{i}+2 x z^{3} \mathbf{j}+y^{4} \mathbf{k} $$

Step-by-Step Solution

Answer

Curl: \( \left( 4y^3 - 6xz^2 \right) \mathbf{i} + 0 \mathbf{j} + (2z^3 - 3x^2) \mathbf{k} \); Divergence: \( 6xy \).

Step 1: Identify Components

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

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

To find the curl, use the formula: \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \mathbf{k} \). Calculate each component: 1. \( \frac{\partial F_3}{\partial y} = 4y^3 \) and \( \frac{\partial F_2}{\partial z} = 6xz^2 \), so the \( i \)-component is \( 4y^3 - 6xz^2 \).2. \( \frac{\partial F_1}{\partial z} = 0 \) and \( \frac{\partial F_3}{\partial x} = 0 \), so the \( j \)-component is \( 0 \).3. \( \frac{\partial F_2}{\partial x} = 2z^3 \) and \( \frac{\partial F_1}{\partial y} = 3x^2 \), so the \( k \)-component is \( 2z^3 - 3x^2 \).Therefore, \( abla \times \mathbf{F} = \left( 4y^3 - 6xz^2 \right) \mathbf{i} + 0 \mathbf{j} + (2z^3 - 3x^2) \mathbf{k} \).

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

The divergence is calculated by \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \). Compute each term:1. \( \frac{\partial F_1}{\partial x} = 6xy \).2. \( \frac{\partial F_2}{\partial y} = 0 \).3. \( \frac{\partial F_3}{\partial z} = 0 \).Sum these to get the divergence: \( abla \cdot \mathbf{F} = 6xy \).

Key Concepts

Understanding Curl in Vector Calculus

In vector calculus, the concept of curl is crucial when analyzing vector fields. The curl of a vector field provides information about the rotational tendency of the field at a particular point. In intuitive terms, you can think of the curl as a measure of how the vector field might "spin" around a point, much like how a whirlpool might rotate water.

Mathematically, the curl is represented by the operator \( abla \times \mathbf{F} \), where \( \mathbf{F} \) is the vector field. It results in another vector field that describes the rotation. Calculation of the curl involves partial derivatives of the vector components and is often expressed in the formula:

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

In the provided solution, the curl of the vector field \( \mathbf{F} \) was found to be \( \left( 4y^3 - 6xz^2 \right) \mathbf{i} + 0 \mathbf{j} + (2z^3 - 3x^2) \mathbf{k} \). This calculation shows us which parts of the field exert rotational tendencies and in what directions, specifically in the \( \mathbf{i} \) and \( \mathbf{k} \) directions in this case.

Divergence: A Measure of Source or Sink Behavior

Divergence is another concept that helps to understand vector fields, especially when discussing how they behave as sources or sinks. Simply put, divergence indicates how the field vectors are spreading out from or converging towards a point. This is often visualized as how a fluid is expanding from or collapsing into regions in the field.

To compute the divergence of a vector field \( \mathbf{F} \), we use the operator \( abla \cdot \mathbf{F} \), which results in a scalar field. This involves the sum of partial derivatives of the vector components:

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

In the given example, the divergence computed was \( 6xy \). This result reveals the rate at which density might increase or decrease at a point, showing that the field likely has a source or sink effect at those points where \( 6xy \) is non-zero.

Introduction to Vector Fields

A vector field is essentially a function that associates a vector to every point in a space. This means that for a given coordinate, the vector field has both a direction and a magnitude defined by components typically labeled \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).

Vector fields are fundamental in physics and engineering as they can represent many physical phenomena, such as wind velocity fields or magnetic and gravitational fields. Each vector in the field points in the direction of the physical force and its length (magnitude) represents the force's strength at that point.

In the exercise, the vector field \( \mathbf{F}(x, y, z)=3 x^{2} y \mathbf{i}+2 x z^{3} \mathbf{j}+y^{4} \mathbf{k} \) consists of three components comprising the field: \( 3x^2 y \), \( 2xz^3 \), and \( y^4 \). Such fields can help visualize complex three-dimensional systems, offering insights into how different physical quantities are distributed across a space.