Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, use Gauss's Divergence Theorem to calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S .\) \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k} ; S\) is the solid enclosed by \(x+y+z=4, x=0, y=0, z=0\).

Step-by-Step Solution

Answer

The flux of \( \mathbf{F} \) across the surface \( \partial S \) is 64.

Step 1: Understanding Gauss's Divergence Theorem

Gauss's Divergence Theorem states that for a vector field \( \mathbf{F} \) defined on a region \( S \) with boundary \( \partial S \), the flux of \( \mathbf{F} \) across \( \partial S \) is equal to the triple integral of the divergence of \( \mathbf{F} \) over \( S \). Mathematically, it is expressed as: \[ \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} (abla \cdot \mathbf{F}) \, dV. \]

Step 2: Compute the Divergence of \( \mathbf{F} \)

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

Step 3: Describe the Solid Region \( S \)

The solid \( S \) is the volume bounded by the planes \( x=0 \), \( y=0 \), \( z=0 \), and \( x+y+z=4 \). This is a tetrahedron with vertices at \( (0,0,0) \), \( (4,0,0) \), \( (0,4,0) \), and \( (0,0,4) \).

Step 4: Set Up the Triple Integral

The limits of integration for the tetrahedron are obtained from the equation \( x+y+z = 4 \). For \( x \), the limits are 0 to 4; for \( y \), the limits are 0 to \( 4-x \); for \( z \), the limits are 0 to \( 4-x-y \). The triple integral becomes: \[ \iiint_{S} (2x + 2y + 2z) \, dV = \int_0^4 \int_0^{4-x} \int_0^{4-x-y} (2x + 2y + 2z) \, dz \, dy \, dx. \]

Step 5: Compute the Integral

First, integrate with respect to \( z \): \[ \int_0^{4-x-y} (2x + 2y + 2z) \, dz = [2xz + 2yz + z^2]_{0}^{4-x-y} = 2x(4-x-y) + 2y(4-x-y) + (4-x-y)^2. \] Simplify and then integrate with respect to \( y \): \[ \int_0^{4-x} [8x - 2x^2 - 2xy + 8y - 2y^2 + (4-x-y)^2] \, dy. \] Finally, integrate with respect to \( x \) from 0 to 4.

Step 6: Evaluate the Integral

After simplifying the expression from Step 5, evaluate the integrals. The step-by-step solution isn't included here for brevity, but you'll find that it evaluates to the volume of the tetrahedron times the average value of \( abla \cdot \mathbf{F} \) over the region \( S \), resulting in a final value of 64.

Key Concepts

Vector Field

A vector field is a map that assigns a vector to every point in a subset of space. Imagine a simple space where each point has a unique set of arrows that indicate direction and magnitude. This is how we can visualize a vector field. For example, in the original exercise, the vector field is given by \(\mathbf{F}(x, y, z) = x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k}\). Here, the vector \((x^2, y^2, z^2)\) is represented at every point \((x, y, z)\) in the space.

The components \(P = x^2\), \(Q = y^2\), and \(R = z^2\) tell us the direction and strength at each point:

• \(P\) influences the \(x\)-axis direction.
• \(Q\) determines the \(y\)-axis direction.
• \(R\) affects the \(z\)-axis direction.

Thus, understanding the vector field provides us a way to analyze various physical phenomena like magnetic fields, fluid flows, and force fields across space.

Divergence

Divergence is a measure of the "spread" of vectors emanating from a point. In a way, you can think of divergence as describing whether more vectors are coming out of a point than going in. Mathematically, for a vector field \(\mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\), the divergence is calculated as:\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]In our exercise, \(\mathbf{F}(x, y, z) = x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k}\), leading to the divergence:\[abla \cdot \mathbf{F} = 2x + 2y + 2z\]This expression helps quantify how much the vector field diverges at any point \((x, y, z)\). A positive divergence implies a net 'outflow' of vectors, like air dispersing from a balloon. Understanding divergence is essential in fluid dynamics, electromagnetism, and more.

Flux

Flux can be understood as the quantity of the vector field passing through a given surface. It represents concepts like the amount of fluid per unit time passing through a surface. When analyzing problems involving vector fields and surfaces, flux helps in understanding how much of the field moves through the surface.

In Gauss's Divergence Theorem, we equate the flux of a vector field across a closed surface to the triple integral of the divergence over the volume inside. This is represented as:\[\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} abla \cdot \mathbf{F} \, dV\]In our specific problem, the surface is the boundary of the solid tetrahedron, and we are interested in finding how much \(\mathbf{F}\) passes through this boundary. Through this theorem, by integrating the divergence, we skip directly calculating the flux and still reach our answer.

Triple Integral

A triple integral extends the idea of an integral to three dimensions, essential for calculating things like volume or total flux across a solid region. Here, the triple integral provides a way to accumulate values (such as divergence) across a solid region in space.

In our Gauss's Divergence solution, we use a triple integral to compute the total of the divergence function over the tetrahedron volume bounded by \(x+y+z = 4\). The limits of integration are detailed such that:

• 0 to 4 defines \(x\).
• 0 to \(4-x\) defines \(y\).
• 0 to \(4-x-y\) defines \(z\).

Thus, our triple integral becomes:\[\iiint_{S} (2x + 2y + 2z) \, dV\]After evaluating the integral step by step, it gives the result as a scalar value. This scalar accounts for the net outflow of the vector field from the enclosed volume, showcasing the practical power of the triple integral in applying Gauss's Divergence Theorem.