Problem 305 - Calculus Volume 3

Question

Compute \(\quad \iint_{S} \mathbf{F} \cdot \mathbf{N} d S,\) where \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+x y z \mathbf{j}+x y z \mathbf{k}\) and \(\mathbf{N}\) is an outward normal vector \(S\), where \(S\) is the surface of the five faces of the unit cube \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\) missing \(z=0\).

Step-by-Step Solution

Answer

The flux through the surface is \( \frac{3}{4} \).

Step 1: Understanding the Problem

We need to compute the flux of the vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + x y z \mathbf{j} + x y z \mathbf{k} \) through the surface \( S \), which is the surface of the unit cube missing the face at \( z = 0 \). This means we have to evaluate the surface integral \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS \) where \( \mathbf{N} \) is the outward normal vector.

Step 2: Applying the Divergence Theorem

Since \( S \) is a closed surface (even though one face is missing, the cube still encloses a volume), we can apply the Divergence Theorem to simplify the computation. According to the Divergence Theorem: \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( V \) is the volume enclosed by \( S \).

Step 3: Calculating the Divergence

Compute the divergence of \( \mathbf{F} \). Since \( \mathbf{F} = x y z \mathbf{i} + x y z \mathbf{j} + x y z \mathbf{k} \), the divergence \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(xyz) + \frac{\partial}{\partial y}(xyz) + \frac{\partial}{\partial z}(xyz) = yz + xz + xy \).

Step 4: Setting Up the Volume Integral

The volume integral becomes \( \iiint_{V} abla \cdot \mathbf{F} \, dV = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} (yz + xz + xy) \, dz \, dy \, dx \). Here, the limits of integration are from 0 to 1 for \( x, y, \) and \( z \) as we are dealing with the unit cube.

Step 5: Evaluating the Volume Integral

Separate the integrals: \( \iiint_{V} abla \cdot \mathbf{F} \, dV = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} yz \, dz \, dy \, dx + \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} xz \, dz \, dy \, dx + \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} xy \, dz \, dy \, dx \). Evaluate each:- First integral \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} yz \, dz \, dy \, dx = \int_{0}^{1} \int_{0}^{1} yz \, (1-0) \, dy \, dx = 1/4 \),- Second integral \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} xz \, dz \, dy \, dx = 1/4 \),- Third integral \( \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} xy \, dz \, dy \, dx = 1/4 \).Sum them up: \( 1/4 + 1/4 + 1/4 = 3/4 \).

Step 6: Adjusting for the Missing Face

Because the face at \( z = 0 \) is missing, the divergence theorem would initially account for the flux through this face, which is zero as no force goes out through it. Thus the result of \( 3/4 \) remains unchanged.

Key Concepts

Flux Calculation

Calculating the flux of a vector field involves determining how much of the vector field flows through a given surface. For a vector field described by \( \mathbf{F} = x y z \mathbf{i} + x y z \mathbf{j} + x y z \mathbf{k} \), we compute the surface integral \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS \), where \( \mathbf{N} \) is the unit outward normal to the surface \( S \).
This integral gives us the total amount of vector field passing through the surface.
Flux calculation is crucial in physics and engineering, helping us understand flow rates across surfaces.

• Calculate \( \mathbf{F} \cdot \mathbf{N} \) at each point on \( S \).
• Integrate this value over the entire surface to find the total flux.
• Consider any missing faces or boundaries, as these adjust the result.

The flux defines how much of, for example, a fluid or electromagnetic field penetrates through \( S \).
Errors often arise from incorrect surface area calculations or normal vector directions.
It's essential to visualize accurately to understand the field's behavior across \( S \).

Vector Fields

Vector fields are functions that assign a vector to every point in space.
In our exercise, the vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + x y z \mathbf{j} + x y z \mathbf{k} \) provides a vector at each point \((x, y, z)\) in the domain.
This field is constant both in direction and magnitude at any point with the same \((x, y, z)\) value.

• Vector fields are crucial for modeling flow systems like fluid dynamics and electromagnetic fields.
• They show both magnitude and direction, which makes them ideal for representing forces.
• The vector \( \mathbf{F} \) represents a force or flow, changing based on the position within the space.

Visualizing vector fields helps to grasp their real-world implications.
Arrows, often larger or smaller, depict the vector magnitude within these fields.
Understanding vector fields ensures grasping potential fields, flow directions, and intensity comprehensively.
Common errors include misunderstandings about vector directions and applying them improperly across boundaries.

Surface Integrals

A surface integral extends the concept of integration to functions defined on a surface.
Essentially, it sums up field components over a curved or flat surface.
In our case, the \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS \) is a surface integral, evaluating the field's effect over \( S \), the cube's surface.

• Surface integrals compute how much of the field crosses a surface, essential in physics for understanding flux.
• These are typically set up as double integrals, addressing the entire surface area.
• The normal vector \( \mathbf{N} \) ensures the integration considers direction accurately.

To calculate, break the area into infinitesimal patches and sum the field values multiplied by the normal vector over the entire surface.
For non-closed surfaces, like the cube lacking \( z=0 \), adjust calculations to compensate for the missing side.
This adjustment aligns the integrals to the actual surface experiencing flux, which may differ from complete enclosures.
Practical concerns involve finding \( \mathbf{N} \) and ensuring the surface area is parameterized correctly.