Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} $$

Step-by-Step Solution

Answer

div \( \mathbf{F} = 2x + 2y + 2z \); curl \( \mathbf{F} = \mathbf{0} \).

Step 1: Understand the Problem

We need to calculate both the divergence and the curl of the vector field \( \mathbf{F}(x, y, z) = x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} \). The divergence gives us a scalar field from a vector field, while the curl gives us another vector field.

Step 2: Apply Divergence Formula

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For our function, \( P = x^2 \), \( Q = y^2 \), and \( R = z^2 \).

Step 3: Calculate Partial Derivatives for Divergence

Find the partial derivatives: \[ \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(x^2) = 2x \] \[ \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(y^2) = 2y \] \[ \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(z^2) = 2z \]

Step 4: Calculate Divergence

Substitute the partial derivatives into the divergence formula: \[ abla \cdot \mathbf{F} = 2x + 2y + 2z \]

Step 5: Apply Curl Formula

The curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\) for our function, \( P = x^2 \), \( Q = y^2 \), and \( R = z^2 \).

Step 6: Calculate Partial Derivatives for Curl

Find the relevant partial derivatives: \[ \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(z^2) = 0 \] \[ \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(y^2) = 0 \] \[ \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(x^2) = 0 \] \[ \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(z^2) = 0 \] \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(y^2) = 0 \] \[ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(x^2) = 0 \]

Step 7: Calculate Curl

Substitute the partial derivatives into the curl formula: \[ abla \times \mathbf{F} = \left( 0 - 0 \right) \mathbf{i} + \left( 0 - 0 \right) \mathbf{j} + \left( 0 - 0 \right) \mathbf{k} = \mathbf{0} \]

Step 8: Final Step: Compile the Results

The divergence of the vector field \( \mathbf{F} \) is \( 2x + 2y + 2z \), and the curl is \( \mathbf{0} \).

Key Concepts

Divergence

The divergence of a vector field is a scalar that gives us an idea of how much the field spreads out or converges at a point. In mathematical terms, it's an operation that measures the magnitude of a vector field's source or sink at a given point. To find the divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), we use the formula: \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \] In our exercise, we have \( P = x^2 \), \( Q = y^2 \), and \( R = z^2 \). By calculating the partial derivatives for each component:

• \( \frac{\partial P}{\partial x} = 2x \)
• \( \frac{\partial Q}{\partial y} = 2y \)
• \( \frac{\partial R}{\partial z} = 2z \)

Substituting these into the divergence formula, we get:\[ abla \cdot \mathbf{F} = 2x + 2y + 2z \] This result tells us that the vector field tends to spread out uniformly across all three axes.

Curl

Curl is another operation in vector calculus. It measures the rotation or twist of a vector field in three-dimensional space. Think of it like the swirl one might see in a whirlpool. Mathematically, the curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is found using the formula:\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \] For our vector field with \( P = x^2 \), \( Q = y^2 \), \( R = z^2 \), we calculate the following partial derivatives:

• \( \frac{\partial R}{\partial y} = 0 \)
• \( \frac{\partial Q}{\partial z} = 0 \)
• \( \frac{\partial P}{\partial z} = 0 \)
• \( \frac{\partial R}{\partial x} = 0 \)
• \( \frac{\partial Q}{\partial x} = 0 \)
• \( \frac{\partial P}{\partial y} = 0 \)

Plugging these values back into the curl formula gives us:\[ abla \times \mathbf{F} = \mathbf{0} \]This simply means our vector field is irrotational, indicating no swirling or twisting at any point in the field.

Partial Derivatives

Partial derivatives are crucial in vector calculus as they allow us to understand how functions change with respect to one variable while keeping other variables constant. They are especially useful when dealing with functions of multiple variables.For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is expressed as \( \frac{\partial f}{\partial x} \). This represents the rate at which the function changes as \( x \) changes, while \( y \) and \( z \) remain constant. Similarly, we have partial derivatives with respect to \( y \) and \( z \).In our vector field \( \mathbf{F}(x, y, z) = x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} \), each component function (like \( x^2 \), \( y^2 \), and \( z^2 \)) was differentiated with respect to its corresponding variable in the calculation of both divergence and curl.

• \( \frac{\partial}{\partial x}(x^2) = 2x \)
• \( \frac{\partial}{\partial y}(y^2) = 2y \)
• \( \frac{\partial}{\partial z}(z^2) = 2z \)

These calculations help determine how each component contributes to the behavior of the vector field. Understanding partial derivatives thus allows us to better interpret results like divergence and curl.