Problem 377 - 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] \(\mathbf{F}(x, y, z)=(\cos y z) \mathbf{i}+e^{x z} \mathbf{j}+3 z^{2} \mathbf{k} ; S\) is the surface of hemisphere \(z=\sqrt{4-x^{2}-y^{2}}\) together with disk \(x^{2}+y^{2} \leq 4\) in the \(x y\) -plane.

Step-by-Step Solution

Answer

The surface integral over \(S\) is \(24\pi\).

Step 1: Identify the Boundary and Enclosed Volume

The given surface, \(S\), is a combination of a hemisphere and the disk in the \(xy\)-plane. Specifically, it is the hemisphere defined by \(z = \sqrt{4-x^{2}-y^{2}}\) and the disk \(x^{2}+y^{2} \leq 4\). The enclosed volume is a complete sphere with radius 2. The divergence theorem can now be applied since it involves a closed surface.

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

The vector field is \(\mathbf{F} = (\cos(yz)) \mathbf{i} + e^{xz} \mathbf{j} + 3z^2 \mathbf{k}\). To find \(abla \cdot \mathbf{F}\), calculate the divergence:\[abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(\cos(yz)) + \frac{\partial}{\partial y}(e^{xz}) + \frac{\partial}{\partial z}(3z^2)\]Evaluating these partial derivatives, we get:\[ 0 + 0 + 6z = 6z\]

Step 3: Apply the Divergence Theorem

The divergence theorem states:\[\int_{S} \mathbf{F} \cdot \mathbf{n} \, ds = \int_{V} (abla \cdot \mathbf{F}) \, dV\]where \(V\) is the volume enclosed by \(S\). With \(abla \cdot \mathbf{F} = 6z\), compute the integral over the volume of the sphere with radius 2 centered at the origin.

Step 4: Set Up the Volume Integral

Convert to spherical coordinates to evaluate the volume integral. In spherical coordinates, points are \((\rho, \theta, \phi)\) with \(\rho\) the radius, \(\theta\) azimuthal angle, and \(\phi\) polar angle. The integration limits for a full sphere are \(0 \leq \rho \leq 2\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq \phi \leq \pi\). The differential volume element is \(\rho^2 \sin \phi \, d\rho \, d\theta \, d\phi\).

Step 5: Evaluate the Volume Integral

The divergence integral becomes:\[\int_{0}^{2} \int_{0}^{2\pi} \int_{0}^{\pi} 6(\rho \cos \phi) \rho^2 \sin \phi \; d\phi \, d\theta \, d\rho\]This evaluates to:\[6 \int_0^2 \rho^3 \, d\rho \cdot \int_0^{2\pi} \; d\theta \cdot \int_0^{\pi} \cos \phi \sin \phi \, d\phi\]Calculate each integral separately:\[\int_0^{\pi} \cos \phi \sin \phi \, d\phi = \frac{1}{2}\]\[\int_0^{2\pi} 1 \, d\theta = 2\pi\]\[\int_0^2 \rho^3 \, d\rho = \frac{16}{4} = 4\]

Step 6: Combine and Simplify Results

Multiply the results from evaluating each integral:\[6 \cdot 4 \cdot 2\pi \cdot \frac{1}{2} = 24\pi\]Thus, the surface integral of \(\mathbf{F}\) over \(S\) is \(24\pi\).

Key Concepts

Surface Integral

A surface integral extends the concept of an integral to dimensions higher than the conventional one. Unlike line integrals, which are defined along a curve, surface integrals operate over surfaces in three-dimensional space. They evaluate how a vector field interacts with a certain surface.

In simpler terms, think of a surface integral as measuring the cumulative effect of a field, like a wind across a large banner. In the context of a vector field, \(\textbf{F}\), and a surface, \S\, the surface integral is represented as \[ \int_{S} \textbf{F} \cdot \textbf{n} \, ds \]where \(\textbf{n}\) is the normal vector to the surface and \(ds\) represents an infinitesimal area element of the surface.

Surface integrals are crucial where the evaluation is needed over a complex surface, such as calculating fluid flow or electromagnetic fields.

Spherical Coordinates

Spherical coordinates provide a powerful method for evaluating integrals in three-dimensional space. These coordinates simplify problems involving symmetry around a point by using \(\rho\), \(\theta\), and \(\phi\):

• \(\rho\) is the radial distance from the origin.
• \(\theta\) is the azimuthal angle in the \(xy\)-plane from the x-axis.
• \(\phi\) is the polar angle from the positive z-axis.

In essence, spherical coordinates resemble how one measures a point's location in a spherical grid, which is particularly advantageous when dealing with spherical or hemispherical regions.

The volume element, or differential volume, in spherical coordinates is given by \( \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi \), allowing us to efficiently evaluate volume integrals by accounting for the volume in terms of these symmetric parameters.

Vector Field Divergence

The concept of divergence in vector fields reflects how much a vector field spreads out from a point. It's a scalar representation of vector fields, indicating how a field like water or air is diverging from or converging to a point.

Given a vector field \(\mathbf{F}(x, y, z) = (F_1, F_2, F_3)\), its divergence is calculated using partial derivatives:\[ abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \]In our specific example, the vector field is \(\mathbf{F}(x, y, z) = (\cos(yz) \,\mathbf{i} + e^{xz} \,\mathbf{j} + 3z^2 \,\mathbf{k})\), yielding a divergence calculation of \(6z\).

Understanding divergence is essential, especially while using the Divergence Theorem, as it relates a surface integral over a closed surface to a volume integral over the region it encloses.

Hemisphere and Disk Surface

A hemisphere and disk surface setup offers a practical visual for understanding 3D surfaces and their enclosed volumes. A hemisphere is essentially half a sphere, typically involved in defining problems of symmetry and integration over curved surfaces.

For this problem, the hemisphere is defined by the equation \(z = \sqrt{4 - x^2 - y^2}\). This equation describes half of a sphere (a cap), where \(z\) is positive.

In combination with the flat disk \(x^2 + y^2 \leq 4\) lying on the \(xy\)-plane, the two create a closed surface when combined, resembling the top half of a ball resting on a flat surface. By analyzing both using surface integrals, we calculate field interactions with these complex surfaces. Integrating over such well-defined surfaces aids in applying the Divergence Theorem effectively.