Problem 389 - Calculus Volume 3

Question

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral \(\int_{S} \mathbf{F} \cdot \mathbf{n} d s\) for the given choice of \(\mathbf{F}\) and the boundary surface \(S\). For each closed surface, assume \(\mathbf{N}\) is the outward unit normal vector. [T] Use a CAS and the divergence theorem to calculate \(\quad\) flux \(\quad \iint_{S} \mathbf{F} \cdot d \mathbf{S}, \quad\) where \(\mathbf{F}(x, y, z)=\left(x^{3}+y^{3}\right) \mathbf{i}+\left(y^{3}+z^{3}\right) \mathbf{j}+\left(z^{3}+x^{3}\right) \mathbf{k}\) and \(S\) is a sphere with center (0,0) and radius 2 .

Step-by-Step Solution

Answer

The flux across the surface is \(\frac{384\pi}{5}\).

Step 1: Understand the Divergence Theorem

The Divergence Theorem, also known as Gauss's Theorem, relates a flux integral over a closed surface to the divergence of a vector field over the volume it encloses. The theorem states: \[ \iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \] where \( V \) is the volume enclosed by surface \( S \).

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

Given \( \mathbf{F}(x,y,z) = (x^3 + y^3)\mathbf{i} + (y^3 + z^3)\mathbf{j} + (z^3 + x^3)\mathbf{k} \), we find the divergence \( abla \cdot \mathbf{F} \). The divergence is:\[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^3 + y^3) + \frac{\partial}{\partial y}(y^3 + z^3) + \frac{\partial}{\partial z}(z^3 + x^3) \] which simplifies to \[ 3x^2 + 3y^2 + 3z^2 \].

Step 3: Set Up the Volume Integral

The goal is to evaluate the integral \( \iiint_{V} (3x^2 + 3y^2 + 3z^2) \ dV \) over the volume of the sphere with radius 2. The sphere of radius 2 centered at the origin can be described in spherical coordinates as \(\rho \leq 2\), \(0 \leq \theta \leq 2\pi\), \(0 \leq \phi \leq \pi\).

Step 4: Convert to Spherical Coordinates

In spherical coordinates, \( x = \rho \sin \phi \cos \theta \), \( y = \rho \sin \phi \sin \theta \), \( z = \rho \cos \phi \). Thus, \( x^2 + y^2 + z^2 = \rho^2 \). The divergence becomes \( 3\rho^2 \).

Step 5: Integrate Over the Volume of the Sphere

The volume integral in spherical coordinates is \[ \iiint_{V} 3\rho^2 \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi = \int_0^{2\pi} \int_0^{\pi} \int_0^2 3\rho^4 \sin \, \phi \, d\rho \, d\phi \, d\theta \].

Step 6: Evaluate the Integrals

First, evaluate the integral with respect to \(\rho\): \[ \int_0^2 3\rho^4 \ d\rho = 3 \left[ \frac{\rho^5}{5} \right]_0^2 = 3 \times \frac{32}{5} = \frac{96}{5} \]. Next, integrate with respect to \(\phi\): \[ \int_0^{\pi} \sin \phi \, d\phi = \left[-\cos(\phi)\right]_0^{\pi} = 2 \]. Finally, integrate the constant with respect to \(\theta\): \[ \int_0^{2\pi} d\theta = 2\pi \].

Step 7: Combine Results

Multiply all components of the integration: \[ \frac{96}{5} \times 2 \times 2\pi = \frac{384\pi}{5} \].

Key Concepts

Surface Integral

A surface integral extends the concept of a regular integral to two dimensions. It is used to calculate the flow of a vector field across a surface. Imagine a surface as a curved sheet in space; the surface integral helps us find how much of the vector field penetrates this sheet.
For a vector field \( \mathbf{F} \), a surface integral over a surface \( S \) is denoted as \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, ds \). Here, \( \mathbf{n} \) represents the outward normal unit vector to the surface.
In simpler terms, the surface integral captures the total 'flow' through the surface. The "outward normal" ensures we measure this flow as it leaves the enclosed volume.

Flux Calculation

Flux calculation involves finding how much of a quantity like fluid or electricity is moving through a surface. In vector calculus, flux is realized through surface integrals.
When utilizing the divergence theorem, it equates the surface integral of the flux through a closed surface \( S \) to the divergence over the volume \( V \) enclosed by \( S \). This converts a 2D problem (the surface) into a simpler 3D problem (the volume) by evaluating:

• The divergence \( abla \cdot \mathbf{F} \) over the volume
• Integrating it to find the total flux

Such computation is beneficial in physics for understanding systems like electromagnetic fields or fluid dynamics.

Spherical Coordinates

Spherical coordinates are a system that extends polar coordinates into three dimensions. They are well-suited for problems involving spheres, as they simplify the description and computation.
In spherical coordinates, every point in space is represented by three numbers: \( \rho \), the radial distance; \( \phi \), the polar angle measured from the positive \( z \)-axis; and \( \theta \), the azimuthal angle in the \( xy \)-plane from the positive \( x \)-axis. Under this system:

• \( x = \rho \sin \phi \cos \theta \)
• \( y = \rho \sin \phi \sin \theta \)
• \( z = \rho \cos \phi \)

This makes it easier to evaluate integrals over spherical volumes, such as in the exercise of finding flux through a sphere.

Vector Calculus

Vector calculus involves differential and integral calculus extended to vector fields. It's essential for operations like finding the divergence or curl of a field.
In the context of the Divergence Theorem, vector calculus allows us to link surface fluxes with volume integrals via the divergence operator. The divergence gives a scalar field representing the rate of change of density of the outward flux of a vector field at a point.
Given a vector field \( \mathbf{F}(x, y, z) \), the divergence \( abla \cdot \mathbf{F} \) is computed using partial derivatives like:\[ abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \]Where \(F_1, F_2,\) and \(F_3\) are the components of \(\mathbf{F}\). This is a vital calculation step in solving the exercise using the Divergence Theorem.