Problem 13 - Calculus: Early Transcendental Functions

Question

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). \(Q \quad\) is bounded by \(\quad x^{2}+y^{2}=1, z=0 \quad\) and \(\quad z=1\) \(\mathbf{F}=\left\langle x-y^{3}, x^{2} \sin z, 3 z\right\rangle\)

Step-by-Step Solution

Answer

The volume integral evaluates to \(2\pi + \pi/2\), so \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S = 2\pi + \pi/2.\)

Step 1: Find the Divergence of the Vector Field

The divergence of the vector field \(\mathbf{F}=\left\langle x-y^{3}, x^{2} \sin z, 3 z\right\rangle\) is computed using the formula div\(\mathbf{F}= \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z} = 1 - 3y^{2} + 3.\)

Step 2: Set Up the Volume Integral

By the Divergence Theorem, the surface integral is equivalent to the volume integral \(\iiint_{O}(\nabla \cdot \mathbf{F})dV\), where the volume \(O\) is bounded by \(x^{2} + y^{2} \le 1\) and \(0 \le z \le 1\). Note that the integral is over a cylinder of radius 1, height 1, which can be conveniently described in cylindrical coordinates \((\rho, \phi, z)\). The volume element in cylindrical coordinates is \(dV = \rho d\rho d\phi dz\). So, the volume integral becomes \(\iiint_{O} \rho(\nabla \cdot \mathbf{F}) d\rho d\phi dz\), where \(0 \le \rho \le 1\), \(0 \le \phi \le 2\pi\), and \(0 \le z \le 1\).

Step 3: Evaluate the Volume Integral

Substitute the divergence \((1 - 3y^{2} + 3)\) of the vector field into the volume integral and convert \(y\) to \(\rho\sin{\phi}\) since we're using cylindrical coordinates: \(\int_{0}^{1}\int_{0}^{2\pi}\int_{0}^{1} \rho(1 - 3(\rho\sin{\phi})^{2} + 3) d\rho d\phi dz\). Now evaluate this triple integral by integrating term by term. The integral of each term is straightforward and yields a real number.

Key Concepts

Vector Calculus

Vector calculus is an essential branch of mathematics, playing a massive role in fields like physics and engineering. It deals with vector fields and operations. In this context:

• Vector fields assign a vector to every point in a subset of space. For example, our field \( \mathbf{F} = \langle x-y^{3}, x^{2} \sin z, 3z \rangle \) assigns values based on the points \( (x, y, z) \).
• Divergence measures a vector field's tendency to originate from or converge at a point. Computed as \( abla \cdot \mathbf{F} = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z} \).

It's crucial here because the Divergence Theorem links the divergence of a vector field in a volume to the field's behavior on the volume's boundary.

Cylindrical Coordinates

Cylindrical coordinates offer a practical way to represent points in three dimensions, especially for symmetrical shapes like cylinders. This coordinate system consists of:

• \( \rho \), the radial distance from the z-axis.
• \( \phi \), the angular coordinate, swept from the x-axis.
• \( z \), the same as Cartesian z-coordinate.

This is extremely useful for expressing volumes and surfaces more simply. The transformation from Cartesian to cylindrical coordinates is defined by:

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

This makes integration over cylindrical shapes straightforward, as seen while setting up the volume integral for our problem.

Volume Integral

A volume integral allows integration over a 3D space, taking the form \( \iiint_{O} f(x, y, z) \, dV \). It sums up quantities inside a volume. In the context of the Divergence Theorem:

• The volume integral \( \iiint_{O} (abla \cdot \mathbf{F}) \, dV \) represents the total divergence inside the volume \( O \).
• For regions like cylinders, cylindrical coordinates simplify computation, turning the volume element into \( \rho \, d\rho \, d\phi \, dz \).
• The integral bounds are specified by the physical dimensions of our volume, \( \rho \) from 0 to 1, \( \phi \) from 0 to \( 2\pi \), and \( z \) from 0 to 1.

Evaluating this integral involves considering the behavior across the entire volume and is crucial in applying the Divergence Theorem effectively.

Surface Integral

A surface integral is a vital tool in calculating fluid flow or electromagnetic effects across surfaces. In general terms:

• The surface integral is represented as \( \iint_{\partial O} \mathbf{F} \cdot \mathbf{n} \, dS \).
• It's a measure of a vector field across a surface, where \( \mathbf{n} \) denotes the unit normal to the surface.

The Divergence Theorem tells us that this surface integral can also be expressed through a volume integral, bridging the field's behavior inside and on the boundary of the volume. This connection often simplifies complex surface integrals by shifting the problem to computing a potentially simpler volume integral, as demonstrated in the provided exercise.