Problem 404 - Calculus Volume 3

Question

For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions \(D\). Let \(\mathbf{F}(x, y, z)=2 x \mathbf{i}-3 y \mathbf{j}+5 z \mathbf{k}\) and let \(S\) be hemisphere \(z=\sqrt{9-x^{2}-y^{2}}\) together with disk \(x^{2}+y^{2} \leq 9\) in the \(x y\) -plane, Use the divergence theorem.

Step-by-Step Solution

Answer

The net outward flux is \(72\pi\).

Step 1: Understand the Vector Field and Region

You have a vector field \(\mathbf{F}(x, y, z) = 2x \mathbf{i} - 3y \mathbf{j} + 5z \mathbf{k}\) and the region \(S\) consists of the hemisphere \(z = \sqrt{9 - x^2 - y^2}\) together with the disk \(x^2 + y^2 \leq 9\) in the \(xy\)-plane. The divergence theorem relates the flux of a vector field through a closed surface to a volume integral over the region bounded by the surface.

Step 2: Calculate the Divergence of the Vector Field

The divergence of the vector field \(\mathbf{F}(x, y, z) = 2x \mathbf{i} - 3y \mathbf{j} + 5z \mathbf{k}\) is computed as \(abla \cdot \mathbf{F} = \frac{\partial}{\partial x} (2x) + \frac{\partial}{\partial y} (-3y) + \frac{\partial}{\partial z} (5z)\). Calculating each component, we get: \(\frac{\partial}{\partial x}(2x) = 2\), \(\frac{\partial}{\partial y}(-3y) = -3\), and \(\frac{\partial}{\partial z}(5z) = 5\). Thus, the divergence is \(abla \cdot \mathbf{F} = 2 - 3 + 5 = 4\).

Step 3: Set Up the Divergence Theorem

According to the divergence theorem, the flux through the closed surface \(S\) is given by the volume integral \(\iiint_{D} (abla \cdot \mathbf{F}) \, dV\), where \(D\) is the volume enclosed by \(S\). Here, \(abla \cdot \mathbf{F} = 4\). The integral becomes \(\iiint_{D} 4 \, dV\).

Step 4: Define the Volume Element and Limits

The region \(D\), a hemisphere, can be effectively described in spherical coordinates. Let \(\rho\) represent the radius from the origin (ranging from 0 to 3), \(\phi\) the angle from the positive z-axis (ranging from 0 to \(\frac{\pi}{2}\)), and \(\theta\) the azimuthal angle (ranging from 0 to \(2\pi\)). The volume element in spherical coordinates is \(dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta\).

Step 5: Compute the Volume Integral

Substitute \(abla \cdot \mathbf{F} = 4\) and integrate over the hemisphere: \[ \iiint_{D} 4 \, dV = \int_{0}^{2\pi} \int_{0}^{\frac{\pi}{2}} \int_{0}^{3} 4 \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta. \]The integration steps are:- \(\int_{0}^{3} 4 \rho^2 \, d\rho = \left[ \frac{4\rho^3}{3} \right]_0^3 = 36\),- \(\int_{0}^{\frac{\pi}{2}} \sin \phi \, d\phi = 1\),- \(\int_{0}^{2\pi} d\theta = 2\pi\).Multiplying the results: \(36 \cdot 1 \cdot 2\pi = 72\pi\).

Step 6: Conclusion

The net outward flux of the vector field \(\mathbf{F}\) across the surface \(S\) is \(72\pi\).

Key Concepts

Vector Field

A vector field is essentially a function that assigns a vector to every point in space. In simpler terms, it's like having a field of arrows where each arrow has a direction and a magnitude. Imagine each point in space having its own arrow, pointing in the direction of the field's influence and stretching in length according to the vector's magnitude.
In the given exercise, the vector field is represented by \(\mathbf{F}(x, y, z) = 2x \mathbf{i} - 3y \mathbf{j} + 5z \mathbf{k}\). Each component — \(2x\), \(-3y\), and \(5z\) — expresses how the field behaves in x, y, and z directions.
This understanding is crucial as it allows us to evaluate how the vector field interacts with different geometrical shapes, like the hemisphere and disk in this problem.

Spherical Coordinates

Spherical coordinates provide a way to define locations in three-dimensional space differently from the usual Cartesian coordinates (x, y, z). They are particularly useful for shapes that are symmetrical around a point, such as spheres.
The system uses three parameters:

• \(\rho\): the distance from the origin.
• \(\phi\): the angle from the positive z-axis.
• \(\theta\): the azimuthal angle, which rotates around the z-axis.

In this exercise, spherical coordinates help us describe the three-dimensional region \(D\) enclosed by the hemisphere. With
\(\rho\) ranging from 0 to 3, \(\phi\) from 0 to \(\frac{\pi}{2}\), and \(\theta\) from 0 to \(2\pi\), we can effectively perform calculations on regions of spherical symmetry, simplifying integration that might otherwise be complex.

Volume Integral

A volume integral, denoted as \(\iiint_{D} f(x, y, z) \, dV\), calculates the total value of a function across a given volume \(D\). Think of it like accumulating the results of the function at each tiny subdivided portion of the volume, then summing it all up.
In the context of the Divergence Theorem, the volume integral serves a special purpose. It allows us to compute the total divergence within the volume enclosed by a surface, which is crucial for determining the flux through that surface.
For the provided exercise, where \( abla \cdot \mathbf{F} = 4 \), we solve the volume integral
\( \iiint_{D} 4 \, dV \).
The integration over \(\rho\), \(\phi\), and \(\theta\) calculates the total value contributed by the vector field's divergence in the region \(D\).

Flux Calculation

Flux represents the amount of a vector field passing through a surface. Visualize this as how much of the vector field's influence or flow cuts across a specified boundary.
In this exercise, we are interested in the 'net outward flux'. This refers to the total flow out of the region \(S\), which is determined via surface integrals linked to the Divergence Theorem.
Rather than computing a potentially complex surface integral, the Divergence Theorem relates it to a simpler volume integral over the enclosed volume \(D\).

• By computing \( \iiint_{D} 4 \, dV \) instead, we harness the theorem to simplify our work.
• The final calculation reveals that the net outward flux through the surface \(S\) is \(72\pi\), showcasing the power of divergence and integration in determining complex field behaviors.

Understanding flux calculation not only aids in solving mathematical problems but also in grasping physical concepts like fluid flow and electromagnetic fields.