Problem 16 - Calculus: Early Transcendental Functions

Question

Use the Divergence Theorem to compute \(\iint_{\partial O} \mathbf{F} \cdot \mathbf{n} d S\). $$\begin{aligned} &\begin{array}{ccccccc} Q & \text { is } & \text { bounded } & \text { by } & z=-\sqrt{4-x^{2}-y^{2}} & \text { and } & z=0 \end{array}\\\ &\mathbf{F}=\left\langle x^{3}, y^{3}, z^{3}\right\rangle \end{aligned}$$

Step-by-Step Solution

Answer

The flux of the vector field \(\mathbf{F}\) across the boundary \(\partial O\) is given by the flux integral is \(\frac{3}{5}\pi\).

Step 1: Calculation of Divergence of the Vector Field

Calculate the divergence of the vector field \(F\), also known as 'del dot F'. In Cartesian coordinates, this is given by the partial derivative of \(x^3\) with respect to \(x\), plus the partial derivative of \(y^3\) with respect to \(y\), plus the partial derivative of \(z^3\) with respect to \(z\). Mathematically it is represented as: div \(\mathbf{F} = 3x^2 + 3y^2 + 3z^2\).

Step 2: Transformation to Spherical Coordinates

Converting the divergence from Cartesian to spherical coordinates simplifies the integral computation. In spherical coordinates, \(x = r\sin(\theta)\cos(\varphi)\), \(y = r\sin(\theta)\sin(\varphi)\), and \(z = r\cos(\theta)\), this gives, div \(\mathbf{F} = 3r^2\sin^2(\theta)\).

Step 3: Setting up the Integral

We will set up a triple integral to compute the volume integral of the divergence over the region. Since the region is bounded by \(z = 0\), \(z = -\sqrt{4 - x^2 - y^2}\), \(\cos(\theta) = 0\) and \(\cos(\theta) = -1\). The triple integral is: \(\int_0^{2\pi}\int_0^{\pi/2}\int_0^2 (3r^2\sin^2(\theta)) r^2\sin(\theta) dr d\theta d\varphi\).

Step 4: Evalute the Integral

On solving the above triple integral, we get \(\frac{3}{5}\pi\) as the flux for divergence of F across boundary \(\partial O\).

Key Concepts

Vector Field

A vector field is a mathematical construct that assigns a vector to every point in a subset of space. In simpler terms, you can think of it as a function that expresses both a magnitude and a direction at each point in the space considered.
For real-world applications, vector fields can represent various quantities like velocity of a fluid in motion or the magnetic force in a region. They are fundamental in physics and engineering, playing a key role in understanding forces and fields.

• For example, the flow of a river can be represented with vectors while in motion and would change as you move from one point of the river to another.
• Similarly, the gravitational forces around a planet can be observed as a vector field with arrows pointing towards the planet's center.

Understanding vector fields well is crucial when dealing with operations like divergence, which measures how much a vector field spreads or converges from a point.

Spherical Coordinates

Spherical coordinates are an alternative to Cartesian coordinates (i.e., the usual 'x, y, z' axes) which can simplify certain types of problems, especially when the region of interest has spherical symmetry.
In spherical coordinates, the position of a point in space is described by three variables:

• \( r \): The radial distance from the origin to the point.
• \( \theta \): The polar angle measured from the positive z-axis.
• \( \varphi \): The azimuthal angle measured in the x-y plane from the positive x-axis.

By converting problems into spherical coordinates, calculations often become more manageable. For example, finding the volume of a sphere or regions bounded by spherical surfaces is typically more straightforward.
In the original exercise, the conversion to spherical coordinates allowed the integral to be simpler by taking advantage of the symmetry of the region defined by the bounds. This conversion is key when dealing with problems involving symmetric limits or spherical domains.

Triple Integral

A triple integral extends the concept of a double integral, allowing you to integrate over a three-dimensional region. This can be used to calculate various properties such as volume, mass, or, as in our exercise, the flux across a surface using the Divergence Theorem.
In essence, a triple integral involves integrating over three variables. These three integrations often correspond to the directions of three axes in a 3D space (like x, y, z, or their spherical counterparts).

• The setup of the integral is crucial. The limits of integration must accurately represent the region you are evaluating over.
• Evaluate each integral step by step. Sometimes symmetry or geometric properties of the region can simplify the process considerably.

In spherical coordinates, when integrating, the differential volume element is given by \( r^2 \sin \theta \, dr \, d\theta \, d\varphi \), as opposed to \( dx \, dy \, dz \) in Cartesian coordinates.
Thus, in the exercise outlined, the triple integral was used to calculate the volume integral over a specific region. Solving such an integral required applying the limits correctly and simplifying wherever possible using spherical coordinates.