Problem 229 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\). \(\mathbf{F}(x, y, z)=a x \mathbf{i}+b y \mathbf{j}+c \mathbf{k}\) for constants \(a, b, c\)

Step-by-Step Solution

Answer

The divergence of \( \mathbf{F} \) is \( a + b \).

Step 1: Write the formula for divergence

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

Step 2: Differentiate with respect to each variable

Calculate \( \frac{\partial P}{\partial x} \), \( \frac{\partial Q}{\partial y} \), and \( \frac{\partial R}{\partial z} \):\[\frac{\partial P}{\partial x} = \frac{\partial (ax)}{\partial x} = a, \\frac{\partial Q}{\partial y} = \frac{\partial (by)}{\partial y} = b, \\frac{\partial R}{\partial z} = \frac{\partial c}{\partial z} = 0.\]

Step 3: Sum the partial derivatives

Now, sum the results of the partial derivatives:\[abla \cdot \mathbf{F} = a + b + 0 = a + b.\]

Key Concepts

Vector Field

A vector field is a fundamental concept in mathematics and physics. It is essentially a function that assigns a vector to each point in space. This can be imagined as an array of arrows, each one representing the vector at a given point. The direction and magnitude of these vectors can change from point to point.In our exercise, the vector field is described by the function \(\mathbf{F}(x, y, z)=a x \mathbf{i}+b y \mathbf{j}+c \mathbf{k}\)where \( a, b, \) and \( c \) are constants. The vector field here involves three-dimensional space, and at any point \( (x, y, z) \), the vector is determined by multiplying the constants with the components \( x, y, \) and \( z \).

• \( a x \mathbf{i} \) contributes to the vector in the x-direction.
• \( b y \mathbf{j} \) contributes in the y-direction.
• \( c \mathbf{k} \) is a constant vector in the z-direction.

Understanding the vector field helps in visualizing forces, fluid flows, and other physical phenomena in a three-dimensional space.

Partial Derivatives

Partial derivatives are a crucial concept, especially when dealing with multivariable functions like vector fields. They measure how a function changes as one of its input variables is varied, keeping the other variables constant. This is incredibly useful when determining rates of change in systems that depend on several variables.For the vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), each component \( P, Q, \) and \( R \) can be functions of \( x, y, \) and \( z \). To find the divergence, we need to compute the partial derivatives:

• \( \frac{\partial P}{\partial x} \) involves differentiating \( P \) with respect to \( x \) while holding \( y \) and \( z \) constant.
• \( \frac{\partial Q}{\partial y} \) involves differentiating \( Q \) with respect to \( y \) while holding \( x \) and \( z \) constant.
• \( \frac{\partial R}{\partial z} \) involves differentiating \( R \) with respect to \( z \) while keeping \( x \) and \( y \) constant.

In our problem, these derivatives simplify because \( P \), \( Q \), and \( R \) are linear functions or constants.

Divergence Formula

The divergence of a vector field is an important operation that provides a measure of how much a vector field spreads out or converges at a point. This is often used in fields such as fluid dynamics and electromagnetism.To find the divergence of a vector field, you apply the divergence formula:\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]This formula adds up the partial derivatives of the components of the vector field.In the original exercise, the vector field \(\mathbf{F}(x, y, z)=a x \mathbf{i}+b y \mathbf{j}+c \mathbf{k}\)is used. By calculating the respective partial derivatives, we obtained:

• \( \frac{\partial (ax)}{\partial x} = a \)
• \( \frac{\partial (by)}{\partial y} = b \)
• \( \frac{\partial c}{\partial z} = 0 \)

Adding these results gives the divergence as \( a + b \), indicating how the vector field spreads out or compresses around different regions in space.