Problem 2 - Thomas Calculus

Question

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=(x \ln y) \mathbf{i}+(y \ln z) \mathbf{j}+(z \ln x) \mathbf{k} $$

Step-by-Step Solution

Answer

The divergence of the field is \( \ln(xyz) \).

Step 1: Understand the Divergence Formula

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

Step 2: Compute Partial Derivative of P with Respect to x

To find \( \frac{\partial P}{\partial x} \), differentiate \( P = x \ln y \) with respect to \( x \), treating \( y \) as a constant. This yields \( \frac{\partial (x \ln y)}{\partial x} = \ln y \).

Step 3: Compute Partial Derivative of Q with Respect to y

To find \( \frac{\partial Q}{\partial y} \), differentiate \( Q = y \ln z \) with respect to \( y \), treating \( z \) as a constant. This yields \( \frac{\partial (y \ln z)}{\partial y} = \ln z \).

Step 4: Compute Partial Derivative of R with Respect to z

To find \( \frac{\partial R}{\partial z} \), differentiate \( R = z \ln x \) with respect to \( z \), treating \( x \) as a constant. This yields \( \frac{\partial (z \ln x)}{\partial z} = \ln x \).

Step 5: Add the Partial Derivatives

Combine the partial derivatives computed in the previous steps: \( abla \cdot \mathbf{F} = \ln y + \ln z + \ln x = \ln(xyz) \). Thus, the divergence of the field is \( \ln(xyz) \).

Key Concepts

Vector Field

In the realm of mathematics, particularly vector calculus, a vector field is a construction where each point in space is associated with a vector. This is a crucial concept in physical sciences and engineering, often representing quantities like velocity fields in fluid dynamics or electromagnetic fields.
These fields are denoted as \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), where \( P \), \( Q \), and \( R \) are functions of the coordinates \( x \), \( y \), and \( z \).

• **Components**: Each vector field consists of three components, which describe the magnitude and direction in each Cartesian coordinate.
• **Representation**: In the example, \( \mathbf{F} = (x \ln y) \mathbf{i} + (y \ln z) \mathbf{j} + (z \ln x) \mathbf{k} \) describes a field where each spatial point determines a vector based on these component functions.

Understanding vector fields is pivotal to grasping phenomena like divergence, which measures how much a vector field spreads out from a point.

Partial Derivatives

Partial derivatives extend the concept of a derivative to functions of multiple variables, where you differentiate with respect to one variable while keeping the others constant.
In vector fields, this is essential for calculating properties like divergence or curl. Let's break down their significance and calculation.

• **Definition**: A partial derivative of a function with several variables, such as \( f(x, y, z) \), is found by differentiating with respect to one variable and treating others as constants.
• **Calculation**: For the field \( \mathbf{F} \), partial derivatives are calculated as follows:
  • For \( P = x \ln y \), the partial derivative with respect to \( x \) is \( \frac{\partial P}{\partial x} = \ln y \).
  • For \( Q = y \ln z \), the partial derivative with respect to \( y \) is \( \ln z \).
  • For \( R = z \ln x \), the partial derivative with respect to \( z \) is \( \ln x \).

These are critical in analyzing changes in vector field components, leading to deeper insights into the behavior of fields in space.

Divergence Theorem

The divergence theorem is a fundamental statement in vector calculus, translating a surface integral into a volume integral and vice versa.
It bridges divergence of vector fields with tangible physical properties, like fluid flow and electric fields.

• **Concept**: The divergence of a vector field at a point provides a scalar that indicates whether there is a source (divergence > 0) or a sink (divergence < 0) at that point.
• **Formula**: Given a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), its divergence is \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
• **Application**: By summing up the partial derivatives as seen in the exercise, we arrive at the divergence \( \ln(xyz) \), conveying how the field behaves around each point.

Understanding the divergence theorem allows us to interpret complex vector behaviors and applications in real-world physics scenarios.