Problem 412 - Calculus Volume 3

Question

Use the divergence theorem to calculate the flux of \(\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} \quad\) through sphere \(x^{2}+y^{2}+z^{2}=1\)

Step-by-Step Solution

Answer

The flux of \(\mathbf{F}\) through the sphere is \(\frac{12\pi}{5}\).

Step 1: Understand the Divergence Theorem

The divergence theorem (also known as Gauss's theorem) connects the flux of a vector field through a closed surface to the divergence of the field over the region inside the surface. It is given by:\[ \int_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \int_{V} abla \cdot \mathbf{F} \, dV \]where \(S\) is a closed surface, \(V\) is the volume inside \(S\), \(\mathbf{n}\) is the outward unit normal, and \(abla \cdot \mathbf{F}\) is the divergence of \(\mathbf{F}\).

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

The vector field is \(\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k}\). The divergence \(abla \cdot \mathbf{F}\) is calculated as:\[ abla \cdot \mathbf{F} = \frac{\partial }{\partial x}(x^3) + \frac{\partial }{\partial y}(y^3) + \frac{\partial }{\partial z}(z^3) \]This leads to:\[ abla \cdot \mathbf{F} = 3x^2 + 3y^2 + 3z^2 \]

Step 3: Setup the Integral in the Volume

The sphere \(x^{2}+y^{2}+z^{2}=1\) describes the boundary of the region. Inside the sphere, the volume integral becomes:\[ \int_{V} (3x^2 + 3y^2 + 3z^2) \, dV \]

Step 4: Switch to Spherical Coordinates

To simplify the calculations, switch to spherical coordinates where \(x = \rho \sin\phi \cos\theta\), \(y = \rho \sin\phi \sin\theta\), and \(z = \rho \cos\phi\). The volume element in spherical coordinates is \(\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta\). The limits for \(\rho\) are from 0 to 1, for \(\phi\) from 0 to \(\pi\), and for \(\theta\) from 0 to \(2\pi\).

Step 5: Evaluate the Integral

The integral in spherical coordinates is:\[\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} 3\rho^2 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \]Calculate the \(\rho\) part:\[\int_{0}^{1} 3\rho^4 \, d\rho = 3\left[\frac{\rho^5}{5}\right]_{0}^{1} = \frac{3}{5}\]Combine it with \(\phi\) and \(\theta\) parts:\[\int_{0}^{\pi} \sin\phi \, d\phi = [-\cos\phi]_{0}^{\pi} = 2\]\[\int_{0}^{2\pi} d\theta = 2\pi\]The final result is:\[\frac{3}{5} \times 2 \times 2\pi = \frac{12\pi}{5}\]

Step 6: Conclude with the Flux

Using the Divergence Theorem, the flux of \(\mathbf{F}\) through the sphere is equal to the volume integral calculated in spherical coordinates. Therefore, the total flux is \(\frac{12\pi}{5}\).

Key Concepts

Flux Calculation

Flux calculation is an essential concept in vector calculus, particularly when dealing with vector fields. It involves quantifying the flow of a vector field across a given surface.
When performing a flux calculation, you look at how much of the vector field passes through a specific surface. Usually, this surface is two-dimensional and can be open or closed.

• If the surface is open, the flux can represent the flow of the vector through it.
• If the surface is closed, like a sphere, it oftentimes leads to applications of Gauss's theorem.

Flux measures the net rate at which "stuff" goes out or comes in through the surface. In mathematical terms, it is represented as the surface integral of the field over this area which can be simplified using the Divergence Theorem for closed surfaces.

Spherical Coordinates

Spherical coordinates provide a system for plotting points in three-dimensional space. They are especially useful when working with problems involving symmetry around a point, such as spheres.
These coordinates use three values: radial distance, azimuthal angle, and polar angle.

• \(\rho\) (rho) is the radial distance from the origin to the point.
• \(\theta\) (theta) is the azimuthal angle, similar to longitude, and measures rotation in the x-y plane.
• \(\phi\) (phi) is the polar angle, akin to latitude, and measures rotation from the z-axis.

The transformation from Cartesian to spherical coordinates means you will use relationships like:

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

This coordinate system simplifies integrating over spherical volumes and surfaces, allowing for straightforward evaluations of integrals in problems with radial symmetry.

Vector Fields

In mathematics and physics, vector fields are a way to represent a quantity that has both magnitude and direction at each point in space.
These fields can describe various physical phenomena, such as magnetic fields, fluid flows, or even gravitational attractions.
A vector field in three dimensions is often expressed in terms of its components along the \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) coordinate axes, such as \(\mathbf{F}(x, y, z) = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}\).
Key characteristics include:

• Direction: Which way a vector points at each location.
• Magnitude: The strength of the vector at every point.

Contemplating the behavior of a vector field can reveal insights about continuity, divergence, and curl, which relate to flux calculations and applications of Gauss's theorem.

Gauss's Theorem

Gauss's theorem, also known as the Divergence Theorem, is a pivotal tool in vector calculus that bridges the gap between surface integrals and volume integrals.
The theorem states that the flux of a vector field across a closed surface is equal to the integral of the divergence of that field over the volume enclosed by the surface.
Mathematically, it is expressed as:\[\int_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \int_{V} abla \cdot \mathbf{F} \, dV\]where:

• \(S\) is the closed surface,
• \(V\) is the volume inside \(S\),
• \(\mathbf{n}\) is the outward unit normal vector to the surface,
• \(abla \cdot \mathbf{F}\) is the divergence of the vector field.

Gauss's theorem enables the conversion of complex surface integrals into more manageable volume integrals, especially when dealing with regions symmetric in nature. It underlines fundamental aspects of flux calculations and vector field concepts, making them crucial in fields like electromagnetism and fluid dynamics.