Problem 6 - Advanced Engineering Mathematics

Question

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=x^{2} \mathbf{i}+2 y z \mathbf{j}+4 z^{3} \mathbf{k} ; D \text { the region bounded by the paral- }\\\ &\text { lelepiped defined by } 0 \leq x \leq 1,0 \leq y \leq 2,0 \leq z \leq 3 \end{aligned} $$

Step-by-Step Solution

Answer

The outward flux is 240.

Step 1: Understand the Problem

We need to find the outward flux of vector field \( \mathbf{F} = x^2 \mathbf{i} + 2yz \mathbf{j} + 4z^3 \mathbf{k} \) through the surface \( S \) of the parallelepiped. The divergence theorem will be used to simplify the computation.

Step 2: Apply the Divergence Theorem

The divergence theorem states \( \iint_{S}(\mathbf{F} \cdot \mathbf{n}) dS = \iiint_{D}(abla \cdot \mathbf{F}) dV \). Thus, to find the flux across surface \( S \), we need to compute the divergence of \( \mathbf{F} \) and integrate it over the volume \( D \).

Step 3: Calculate the Divergence of \( \mathbf{F} \)

Find the divergence \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(2yz) + \frac{\partial}{\partial z}(4z^3) \). \( \frac{\partial}{\partial x}(x^2) = 2x \), \( \frac{\partial}{\partial y}(2yz) = 2z \), and \( \frac{\partial}{\partial z}(4z^3) = 12z^2 \). Thus, \( abla \cdot \mathbf{F} = 2x + 2z + 12z^2 \).

Step 4: Set Up the Triple Integral

We integrate \( 2x + 2z + 12z^2 \) over the volume \( D \) defined by \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 2 \), \( 0 \leq z \leq 3 \). The integral becomes: \[ \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} (2x + 2z + 12z^2) \, dz \, dy \, dx \].

Step 5: Evaluate the Inner Integral (with respect to z)

First integrate \( 2x + 2z + 12z^2 \) with respect to \( z \):\[ \int_{0}^{3} (2x + 2z + 12z^2) \, dz = \left[ 2xz + z^2 + 4z^3 \right]_{0}^{3} = 6x + 9 + 108 = 6x + 117 \].

Step 6: Evaluate the Middle Integral (with respect to y)

Integrate \( 6x + 117 \) with respect to \( y \):\[ \int_{0}^{2} (6x + 117) \, dy = \left[ 6xy + 117y \right]_{0}^{2} = 12x + 234 \].

Step 7: Evaluate the Outer Integral (with respect to x)

Finally, integrate \( 12x + 234 \) with respect to \( x \):\[ \int_{0}^{1} (12x + 234) \, dx = \left[ 6x^2 + 234x \right]_{0}^{1} = 6 + 234 = 240 \].Thus, the outward flux through surface \( S \) is 240.

Key Concepts

Outward Flux

Outward flux is a concept in vector calculus that measures how much a vector field moves outward through a surface. When you have a vector field, it's important to know how that field behaves around a surface, like in our case, a parallelepiped. To find the outward flux, we essentially need to calculate how much of the vector field points out of the surface.

This calculation is simplified using the Divergence Theorem. Instead of dealing with a complex surface integral, the theorem allows us to convert it into a volume integral over the region inside the surface. By computing the flux this way, we consider the collective effect of the field within the volume, making it easier to handle mathematically.

In our problem, the formula \[ \iint_{S} (\mathbf{F} \cdot \mathbf{n}) \, dS \] helps identify the outward flux of the vector field \( \mathbf{F} \) through surface \( S \) of the parallelepiped.

Vector Field

A vector field is a function that assigns a vector to each point in space. In simple terms, it can be thought of as a set of arrows scattered throughout a region, where each arrow has both direction and magnitude. In our exercise, the vector field \( \mathbf{F} = x^2 \mathbf{i} + 2yz \mathbf{j} + 4z^3 \mathbf{k} \) describes how forces, velocities, or other vector quantities vary across the space of the parallelepiped.

The components of the vector field determine its behavior. For example:

• \( x^2 \mathbf{i} \) shows how the field varies along the x-axis.
• \( 2yz \mathbf{j} \) reflects changes along the y-axis, dependent on both \( y \) and \( z \).
• \( 4z^3 \mathbf{k} \) intensifies as \( z \) increases, dominating the movement along the z-axis.

Understanding the vector field is crucial because it affects how we apply the Divergence Theorem to compute the flux.

Triple Integral

The triple integral is a way of integrating over three-dimensional regions. In simple terms, think of it as summing up small volumes within a 3D shape, like a parallelepiped in our example. When applying the Divergence Theorem, we use a triple integral to represent the volume integral of the divergence of the vector field.

In our problem, the triple integral \[ \iiint_{D} (abla \cdot \mathbf{F}) \, dV \] is used to compute the total divergence across the volume of the parallelepiped. By breaking down the integration process step-by-step:

• We first integrate with respect to \( z \), which accounts for changes in height within the shape.
• Next, we integrate with respect to \( y \), considering the width of the region.
• Lastly, we integrate with respect to \( x \), covering the length of the parallelepiped.

These calculations lead to the final result, telling us how much the vector field is diverging out of the whole volume.

Parallelepiped

A parallelepiped is a six-faced figure (also known as a polyhedron) where each face is a parallelogram. Imagine it as a box with slanted sides instead of straight up-and-down ones.

In our exercise, the bounded region \( D \) is a parallelepiped defined by the inequalities \( 0 \leq x \leq 1 \), \( 0 \leq y \leq 2 \), and \( 0 \leq z \leq 3 \). These inequalities form the constraints for setting up the limits of our triple integral.

Thinking of a parallelepiped can be helpful when visualizing how the vector field interacts with the boundary as well as how the volume integral is performed inside it. This spatial understanding aids in grasping concepts like flux and divergence, ensuring that the theoretical calculations align with real-world scenarios.