Problem 231 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} $$

Step-by-Step Solution

Answer

The divergence of \(\mathbf{F}\) is \(x + y + z\).

Step 1: Understanding the Divergence Operator

The divergence of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is calculated as the scalar function \(abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\). Our task is to find these partial derivatives for each of the components of \(\mathbf{F}\).

Step 2: Calculate the Partial Derivative with Respect to x

For the vector field \(\mathbf{F}(x, y, z) = x y \mathbf{i} + y z \mathbf{j} + x z \mathbf{k}\), the function \(P(x, y, z) = xy\). Calculate the partial derivative of \(P\) with respect to \(x\), which is \(\frac{\partial}{\partial x}(xy) = y\).

Step 3: Calculate the Partial Derivative with Respect to y

The function \(Q(x, y, z) = yz\). Calculate the partial derivative of \(Q\) with respect to \(y\), which is \(\frac{\partial}{\partial y}(yz) = z\).

Step 4: Calculate the Partial Derivative with Respect to z

The function \(R(x, y, z) = xz\). Calculate the partial derivative of \(R\) with respect to \(z\), which is \(\frac{\partial}{\partial z}(xz) = x\).

Step 5: Sum the Partial Derivatives

Add the partial derivatives obtained: \(y + z + x\). This sum gives the divergence of the vector field \(\mathbf{F}\).

Step 6: Final Calculation and Conclusion

Thus, the divergence of \(\mathbf{F}(x, y, z) = xy \mathbf{i} + yz \mathbf{j} + xz \mathbf{k}\) is \(x + y + z\).

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in space. Picture it like a field of arrows floating in three-dimensional space. Each arrow represents a vector defined by its magnitude and direction. For the vector field \[ \mathbf{F}(x, y, z) = x y \mathbf{i} + y z \mathbf{j} + x z \mathbf{k}, \]each component (\(xy, yz, xz\)) determines the vector's direction at any given point \((x, y, z)\). This creates a system where vectors change their values based on their position in the coordinate space.

• \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors pointing in the directions of the x, y, and z axes respectively.
• The goal is to understand how these vectors behave and interact, which the divergence can help uncover.

These fields are fundamental in physics and engineering, representing forces such as magnetic and electric fields.

Partial Derivatives

Partial derivatives are a way to show how a function changes as one of its variables changes, while keeping other variables constant. They are crucial for understanding the behavior of multivariable functions, like our vector field.
For our vector field \[\mathbf{F}(x, y, z) = x y \mathbf{i} + y z \mathbf{j} + x z \mathbf{k}, \]we compute the partial derivative for each component:

• For \(P(x, y, z) = xy\), the partial derivative with respect to \(x\) is \(y\).
• For \(Q(x, y, z) = yz\), the partial derivative with respect to \(y\) is \(z\).
• For \(R(x, y, z) = xz\), the partial derivative with respect to \(z\) is \(x\).

These derivatives tell us how each component of the vector field changes, giving insights into the field's behavior and helping calculate the divergence.

Divergence Theorem

The Divergence Theorem is a powerful tool in calculus connecting flux through a surface to behavior inside the volume. It states that the total divergence within a volume is equal to the flux across the volume's boundary. This theorem supports our understanding of how vector fields flow through space.
For our vector field, the divergence, calculated as\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]turns into the expression \(x + y + z\).
By calculating divergence, you get a scalar field representing how much the vector field diverges ("spreads out") from each point, which can then be related to the flux through a surface by the Divergence Theorem.

Calculus

Calculus is the branch of mathematics that studies changes and motion, offering tools to analyze and describe the world around us. It includes differential calculus, which examines rates of change, and integral calculus, which studies accumulation and areas.
In the context of this exercise, calculus enables the computation of partial derivatives and divergence. These calculations involve changing one rate while examining its impact within a multidimensional space.

• Partial derivatives show how a function changes with each individual variable, maintaining all others constant.
• Divergence considers the spread of the vector field, integrating changes across all axes to provide a single value that encapsulates behavior at a point.

Through calculus, the complexities of vector fields and their divergences are unraveled, allowing deeper insights into phenomena modeled by these mathematical structures.