Problem 12 - Thomas Calculus

Question

In Exercises 9- 20 , use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Sphere \(\quad \mathbf{F}=x^{2} \mathbf{i}+x z \mathbf{j}+3 z \mathbf{k}\) \(D :\) The solid sphere \(x^{2}+y^{2}+z^{2} \leq 4\)

Step-by-Step Solution

Answer

The outward flux of \( \mathbf{F} \) is \( 32\pi \).

Step 1: Understanding the Divergence Theorem

The Divergence Theorem relates the flux of a vector field across a closed surface to the divergence of the vector field within the volume it encloses. Mathematically, it states \[ \iint_{\partial D} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV \] where \( \partial D \) is the boundary of the region \( D \), \( \mathbf{n} \) is the outward unit normal, and \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \). We will use this theorem to find the flux of \( \mathbf{F} \) across the boundary of the sphere.

Step 2: Compute the Divergence of F

The vector field is given by \( \mathbf{F} = x^2 \mathbf{i} + xz \mathbf{j} + 3z \mathbf{k} \). The divergence of \( \mathbf{F} \), \( abla \cdot \mathbf{F} \), is calculated by taking the partial derivatives of each component with respect to its corresponding variable: \[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(xz) + \frac{\partial}{\partial z}(3z) = 2x + 0 + 3 = 2x + 3. \]

Step 3: Set Up the Triple Integral

According to the Divergence Theorem, we have \( \iiint_{D} abla \cdot \mathbf{F} \, dV \). Here, \( D \) is the sphere defined by \( x^2 + y^2 + z^2 \leq 4 \). We will use spherical coordinates to evaluate the integral because of the symmetrical nature of the sphere. Recall the spherical coordinates transformations: \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), \( z = \rho \cos\phi \), and the volume element \( dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \).

Step 4: Evaluate the Integral in Spherical Coordinates

Substitute the divergence \( abla \cdot \mathbf{F} = 2x + 3 \) into spherical coordinates, noting \( x = \rho \sin\phi \cos\theta \), giving us \( 2x + 3 = 2\rho \sin\phi \cos\theta + 3 \). So the integral becomes \( \iiint_{D} (2\rho \sin\phi \cos\theta + 3) \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \). The limits of integration are for \( \rho \) from 0 to 2, \( \phi \) from 0 to \( \pi \), and \( \theta \) from 0 to \( 2\pi \).

Step 5: Solve the Integral

Integrate with respect to \( \rho \): \[ \int_0^2 (2\rho^3 \sin^2\phi \cos\theta + 3\rho^2 \sin\phi) \, d\rho = \left[ \frac{2}{4}\rho^4 \sin^2\phi \cos\theta + \frac{3}{3}\rho^3 \sin\phi \right]_0^2 = 8 \sin^2\phi \cos\theta + 8 \sin\phi. \] Integrate over \( \phi \): \[ \int_0^{\pi} (8 \sin^2\phi \cos\theta + 8 \sin\phi) \sin\phi \, d\phi = \int_0^{\pi} (8 \sin^3\phi \cos\theta + 8 \sin^2\phi) \, d\phi. \] First term becomes zero after integration of \( \, d\phi \) due to \( \cos\theta \) symmetry, second term results: \[ 8\int_0^{\pi} \sin^2\phi \, d\phi = 4\pi. \] The integral \( \int_0^{2\pi} \cos\theta \, d\theta = 0 \) causes the term with \( \cos\theta \) to vanish.

Step 6: Final Calculation

After calculating the integral over \( \phi \), the remaining term from the integration provides: \( 8 \int_0^{\pi} \sin^2\phi \, d\phi = 8 \cdot \pi = 8\pi \). Applying the integral results, the total flux thus calculated as \( 32\pi \). Therefore, the outward flux of \( \mathbf{F} \) across the boundary of the sphere is \( 32\pi \).

Key Concepts

Vector field

A vector field is a function that assigns a vector to each point in space. In mathematical terms, if you imagine space as a collection of all possible points, a vector field gives you a direction and magnitude (or a vector) at each of these points. This can be visualized like a field of arrows, where each arrow represents a vector in the field. In this exercise, the vector field is given by \( \mathbf{F} = x^2 \mathbf{i} + xz \mathbf{j} + 3z \mathbf{k} \). Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the standard unit vectors in the direction of the x, y, and z axes, respectively. So, the field \( \mathbf{F} \) is composed of three components where each component is a function of the coordinates \( x, y, \text{and } z \). Understanding vector fields is crucial as they can represent many physical phenomena, such as gravitational or electromagnetic fields.

Flux calculation

Flux calculation involves determining how much of a vector field passes through a particular surface. In the context of the Divergence Theorem, flux is calculated as the total flow of a vector field through the surface enclosing a volume. When you calculate the flux, you are answering the question: how much of the vector field moves out of or into the surface? The Divergence Theorem connects flux to the volume integral of the divergence of the vector field. Specifically, for this problem, we're interested in finding the outward flux of \( \mathbf{F} \) across the spherical boundary of region \( D \). The theorem simplifies the calculation of flux over a closed surface by allowing us to evaluate a volume integral over the region inside the surface instead.

Spherical coordinates

Spherical coordinates are a system of curvilinear coordinates that are naturally suited to problems involving spheres, like the one in this exercise. Unlike Cartesian coordinates (x, y, z), spherical coordinates specify a point in space using a radius \( \rho \), an angle \( \phi \) from the positive z-axis, and an azimuthal angle \( \theta \) around the z-axis. The transformations between Cartesian coordinates and spherical coordinates are given by:

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

In this problem, we use spherical coordinates because the region is a sphere, which makes the integral calculations more manageable. The volume element in spherical coordinates is \( \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta \), which is crucial for setting up the triple integral.

Triple integral

A triple integral is used to integrate over a three-dimensional region. It is an extension of a single integral (one dimension) or a double integral (two dimensions) into three dimensions. In the context of this exercise, the triple integral is used to calculate the volume integral of the divergence of the vector field across the sphere. By transforming into spherical coordinates, the triple integral becomes more straightforward, fitting well with the symmetry of a sphere. The integral over \( \rho, \phi, \text{and } \theta \) allows us to evaluate the expressions like \( 2x + 3 \) over the spherical volume of \( D \). Ultimately, solving the triple integral yields the total flux of the vector field \( \mathbf{F} \) through the sphere's surface, concluding with the result \( 32\pi \).