Problem 3 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The divergence of the field is \((y^2 z + x z^2 + x^2 y) e^{x y z}\).

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 \(abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\). In this problem, \(P = y e^{x y z}\), \(Q = z e^{x y z}\), and \(R = x e^{x y z}\).

Step 2: Calculate Partial Derivative \(\frac{\partial P}{\partial x}\)

For \(P = y e^{x y z}\), treat \(y\) as a constant and differentiate with respect to \(x\). Use the chain rule: \(\frac{\partial}{\partial x}(e^{x y z}) = y z e^{x y z}\). Therefore, \(\frac{\partial P}{\partial x} = y^2 z e^{x y z}\).

Step 3: Calculate Partial Derivative \(\frac{\partial Q}{\partial y}\)

For \(Q = z e^{x y z}\), treat \(z\) as a constant and differentiate with respect to \(y\). Use the chain rule: \(\frac{\partial}{\partial y}(e^{x y z}) = x z e^{x y z}\). Therefore, \(\frac{\partial Q}{\partial y} = x z^2 e^{x y z}\).

Step 4: Calculate Partial Derivative \(\frac{\partial R}{\partial z}\)

For \(R = x e^{x y z}\), treat \(x\) as a constant and differentiate with respect to \(z\). Use the chain rule: \(\frac{\partial}{\partial z}(e^{x y z}) = x y e^{x y z}\). Therefore, \(\frac{\partial R}{\partial z} = x^2 y e^{x y z}\).

Step 5: Sum the Partial Derivatives to Find Divergence

Combine the partial derivatives to find the divergence: \(abla \cdot \mathbf{F} = y^2 z e^{x y z} + x z^2 e^{x y z} + x^2 y e^{x y z}\). This simplifies to \((y^2 z + x z^2 + x^2 y) e^{x y z}\).

Key Concepts

Vector Field

In mathematics, a vector field assigns a vector to every point in space. Imagine a vector field as an invisible force guiding particles along certain directions. This can be gravitational fields, electromagnetic fields, or velocity fields of fluids.
The vector field in this exercise is \[ \mathbf{F}=y e^{x y z} \mathbf{i}+z e^{x y z} \mathbf{j}+x e^{x y z} \mathbf{k} \]where each component (\( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \)) represents a different direction in 3D space.
Analyzing the behavior of this field involves finding quantities like divergence, which informs us about the "spread" or "concentration" of the vectors at each point.

Partial Derivatives

Partial derivatives measure how a function changes as only one of the variables changes, with all other variables held constant. In the context of vector fields, they are crucial for understanding how each vector component changes in space.
For the vector field \( \mathbf{F} \), we find the partial derivatives of each component, \( P \), \( Q \), and \( R \), with respect to different variables.

• For \( \frac{\partial P}{\partial x} \), treat \( y \) and \( z \) as constants, and differentiate \( P = y e^{x y z} \).
• For \( \frac{\partial Q}{\partial y} \), treat \( x \) and \( z \) as constants, and differentiate \( Q = z e^{x y z} \).
• For \( \frac{\partial R}{\partial z} \), treat \( x \) and \( y \) as constants, and differentiate \( R = x e^{x y z} \).

This step helps in finding the divergence, a core operation in vector calculus.

Chain Rule

The chain rule is a vital tool in calculus for differentiating composite functions.
It essentially tells you how to handle derivatives when dealing with functions embedded within one another. In this problem, every component of the vector field involves an exponential term, \( e^{x y z} \).
Using the chain rule:

• For \( \frac{\partial}{\partial x}(e^{x y z}) \), the derivative is \( y z e^{x y z} \).
• For \( \frac{\partial}{\partial y}(e^{x y z}) \), the derivative is \( x z e^{x y z} \).
• For \( \frac{\partial}{\partial z}(e^{x y z}) \), the derivative is \( x y e^{x y z} \).

Understanding this rule simplifies the calculations of partial derivatives in composite functions found in vector fields.

Vector Calculus

Vector calculus is the field of mathematics focused on multivariable calculus in vector fields. It involves operations like divergence, gradient, and curl. These tools allow us to describe and analyze physical quantities that vary over space.
For this exercise, we are primarily interested in the divergence, which can be thought of as a measure of the field's tendency to originate from or converge into points.
This involves adding up all partial derivatives of the components of a vector field.
The divergence formula is: \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]
By calculating this, we determine the overall behavior of the vector field, as seen in combining the partial derivatives of the field \( \mathbf{F} \): \[ (y^2 z + x z^2 + x^2 y) e^{x y z} \]
This result tells us about how vectors in the field are distributed in space.