Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, use Gauss's Divergence Theorem to calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S .\) \(\mathbf{F}(x, y, z)=2 x \mathbf{i}+3 y \mathbf{j}+4 z \mathbf{k} ; S\) is the solid spherical shell \(9 \leq x^{2}+y^{2}+z^{2} \leq 25\).

Step-by-Step Solution

Answer

The flux through the surface is \( 1176 \pi \).

Step 1: Understand the Gauss's Divergence Theorem

Gauss's Divergence Theorem states \( \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} abla \cdot \mathbf{F} \, dV \), where \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \). This converts a surface integral over a closed surface \( \partial S \) to a volume integral over the region \( S \) it encloses.

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

For \( \mathbf{F}(x, y, z) = 2x \mathbf{i} + 3y \mathbf{j} + 4z \mathbf{k} \), its divergence is \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(4z) = 2 + 3 + 4 = 9 \).

Step 3: Define the Volume Region \( S \)

The solid \( S \) is a spherical shell defined by \( 9 \leq x^2 + y^2 + z^2 \leq 25 \). This corresponds to a spherical shell with an inner radius of 3 and an outer radius of 5.

Step 4: Set Up the Volume Integral

Since \( abla \cdot \mathbf{F} = 9 \), the integral \( \iiint_{S} abla \cdot \mathbf{F} \ dV = 9 \cdot \text{Volume of } S \). The volume of a spherical shell is given by \( \frac{4}{3}\pi(R^3 - r^3) \) where \( R \) is the outer radius and \( r \) is the inner radius.

Step 5: Calculate the Volume of the Spherical Shell

Here, \( R = 5 \) and \( r = 3 \). Thus, the volume is \( \frac{4}{3}\pi((5)^3 - (3)^3) = \frac{4}{3}\pi(125 - 27) = \frac{4}{3}\pi \cdot 98 = \frac{392}{3}\pi \).

Step 6: Compute the Integral Result

Multiply the divergence result by the volume: \( 9 \cdot \frac{392}{3}\pi = 1176 \pi \). Thus, the flux through the closed surface is \( 1176 \pi \).

Key Concepts

Surface Integral

A surface integral is a way to calculate the total flux of a vector field across a surface. It helps us understand how a vector field interacts with a surface through which it passes.
In mathematics, this often involves calculating how much of the field passes through a surface in a given direction.

In the context of Gauss's Divergence Theorem, the surface integral \[ \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS \] represents the flow of the vector field \( \mathbf{F} \) across a closed surface \( \partial S \). Here, \( \mathbf{n} \) is the unit normal vector pointing out of the surface.

Unlike simpler integrals, surface integrals must consider the orientation of the surface and the direction of the vector field:

• Orientation: It determines the direction that is considered positive for the flux calculation.
• Vector Field: Shows the magnitude and direction in which the field points as it interacts with the surface.

This complexity is what makes surface integrals a powerful tool in physics and engineering for solving problems related to fluid dynamics, electromagnetism, and more.

Volume Integral

Volume integrals help us find the total quantity of a field within a region of space. These integrals are crucial when you need to calculate the amount of something spread across a three-dimensional space.

For Gauss's Divergence Theorem, it's depicted as:
\[ \iiint_{S} abla \cdot \mathbf{F} \, dV \] This notation means we are integrating the divergence of \( \mathbf{F} \) over the volume \( S \) enclosed by the surface \( \partial S \).

Key aspects to remember when considering volume integrals include:

• Divergence: It sums up what happens to the vector field inside the volume, whether the field is acting as a source or sink.
• Enclosed Volume: The space within which we are evaluating the field's total effect.

By computing volume integrals, we can quantify physical properties like mass, charge, or energy within a volume, making it an essential tool in physics and engineering.

Divergence of a Vector Field

The divergence of a vector field is a critical measure that helps us understand how much a vector field spreads out or converges at a given point. It essentially quantifies the 'outflowing-ness' at each point in the field.

To calculate divergence, we take the dot product of the del operator (\(abla \)) with the vector field \( \mathbf{F} \).
For instance, in our exercise, for \( \mathbf{F}(x, y, z) = 2x \mathbf{i} + 3y \mathbf{j} + 4z \mathbf{k} \), the divergence is:
\[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(2x) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(4z) = 2 + 3 + 4 = 9 \]
This shows how much the vector field is acting like a source inside the volume of interest.

Some fundamental aspects of divergence include:

• Positive Divergence: Represents a net flow out of a point, signifying a source.
• Negative Divergence: Indicates a net flow into a point, resembling a sink.
• Zero Divergence: Suggests a balanced or conserved flow.

Understanding divergence is essential in fields like fluid mechanics, electromagnetism, and thermodynamics, where it helps describe the behavior of fields within different environments.