Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The integral evaluates to 8.

Step 1: Write Gauss's Divergence Theorem

Gauss's Divergence Theorem states that \( \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S = \iiint_S (abla \cdot \mathbf{F}) \, dV \), where \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \) and \( S \) is the solid region with boundary \( \partial S \).

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

Given \( \mathbf{F}(x, y, z) = \cos z^2 \mathbf{i} + y \mathbf{j} + \cos x^2 \mathbf{k} \), calculate the divergence: \[abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(\cos z^2) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(\cos x^2)\] This simplifies to \[0 + 1 + 0 = 1.\]

Step 3: Set Up the Volume Integral

The region \( S \) is the cube defined by \(-1 \leq x \leq 1\), \(-1 \leq y \leq 1\), and \(-1 \leq z \leq 1\). Thus, \( \iiint_S 1 \, dV \) becomes the integral of 1 over the volume of the cube.

Step 4: Evaluate the Volume Integral

Since the divergence is 1, and the volume of the cube \( S \) is \((2)(2)(2) = 8\), calculate the integral: \[\iiint_S 1 \, dV = 8.\]

Step 5: Conclude with the Result

By applying Gauss's Divergence Theorem, \( \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S = \iiint_S (abla \cdot \mathbf{F}) \, dV = 8.\)

Key Concepts

Divergence

Divergence is a fundamental concept in vector calculus, playing a pivotal role in Gauss's Divergence Theorem. You can think of it as a measure of how much a vector field spreads out from a point. It’s somewhat like assessing if a fluid is expanding or contracting at different points in space. Divergence is calculated using a mathematical operator called the "del" operator (∇). Given a vector field \( \mathbf{F} \), its divergence is written as \( abla \cdot \mathbf{F} \). In our given problem, the vector field is \( \mathbf{F}(x, y, z) = \cos z^2 \mathbf{i} + y \mathbf{j} + \cos x^2 \mathbf{k} \).
To find its divergence, we take the partial derivatives of each component relative to its variable:

• \( \frac{\partial}{\partial x}(\cos z^2) \) = 0
• \( \frac{\partial}{\partial y}(y) \) = 1
• \( \frac{\partial}{\partial z}(\cos x^2) \) = 0

Adding these results, the divergence is simply \( 1 \). This uniform result indicates a steady expansion throughout the spatial region considered in our cube, a crucial step before finding the volume integral.

Volume Integral

A volume integral is integral calculus that's extended into three dimensions. When calculating the volume integral of a scalar function over a three-dimensional region, you're essentially summing values throughout the space.
In our exercise, since the divergence of the given vector field \( abla \cdot \mathbf{F} \) is 1, the volume integral over the cube becomes relatively simple. We need to compute \( \iiint_S 1 \, dV \), where the region \( S \) is the cube defined by the boundaries \(-1 \leq x \leq 1\), \(-1 \leq y \leq 1\), \( -1 \leq z \leq 1\).
This problem is straightforward because the integrand is constant (1). We calculate the volume of the cube, which is the product of its dimensions in all directions: \((2 \times 2 \times 2) = 8\). This result represents the total volume of the cube, over which the divergence is integrated. Thus, the overall volume integral evaluates to 8, demonstrating the total "spread" of our vector field within the cube.

Vector Calculus

Vector calculus is a broad field of mathematics that deals with vector fields and the operations that can be performed on them. It extends calculus concepts, such as derivatives and integrals, to multivariable situations.
Key operations in vector calculus include:

• Gradient, which relates to scalar functions and provides a vector that points in the direction of maximum increase.
• Divergence, which we've used in this problem to indicate how much a field spreads out.
• Curl, concerning rotational effects in vector fields.

In our context, vector calculus provides the framework needed to apply Gauss's Divergence Theorem, integrating the divergence of a vector field over a volume to find the net flow across the surface boundary of that volume. This mathematical structure and language help physicists and engineers describe real-world phenomena, such as electromagnetism and fluid dynamics, in a precise way.

Cube

A cube in mathematical terms is a three-dimensional geometric shape with all sides and angles equal. In this exercise, the cube is defined by the coordinates \(-1 \leq x \leq 1\), \(-1 \leq y \leq 1\), and \(-1 \leq z \leq 1\).
This representation means each side of the cube spans 2 units in space along each axis. Essential properties of a cube include:

• Six faces, each a square with equal area.
• Twelve edges, all of the same length.
• Eight vertices where the edges meet.

The uniformity and symmetry of a cube make it an ideal shape for applying theories like Gauss's Divergence Theorem, as its regular geometry simplifies the integration processes. In this example, calculating a volume integral over a cube involves straightforward bounds that contribute to the ease of computing the desired results.