Problem 7 - Advanced Engineering Mathematics

Question

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=y^{3} \mathbf{i}+x^{3} \mathbf{j}+z^{3} \mathbf{k} ; D \text { the region bounded within by }\\\ &z=\sqrt{4-x^{2}-y^{2}}, x^{2}+y^{2}=3, z=0 \end{aligned} $$

Step-by-Step Solution

Answer

Determine \( \iiint_{D} 3z^2 \, dV \) using cylindrical coordinates to find the flux.

Step 1: Understand the Problem Statement

We are tasked to find the outward flux of the vector field \( \mathbf{F}=y^3 \mathbf{i} + x^3 \mathbf{j} + z^3 \mathbf{k} \) across the surface \( S \) using the divergence theorem. The region \( D \) is bounded by the surfaces \( z = \sqrt{4-x^2-y^2} \), \( x^2+y^2 = 3 \), and \( z = 0 \).

Step 2: Express Divergence Theorem

According to the divergence theorem, the surface integral of a vector field over a surface \( S \) can be converted into a triple integral over the volume \( D \) enclosed by the surface: \[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV. \] We need to compute \( abla \cdot \mathbf{F} \) first.

Step 3: Calculate the Divergence

The divergence of \( \mathbf{F} \) is calculated as \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(y^3) + \frac{\partial}{\partial y}(x^3) + \frac{\partial}{\partial z}(z^3) \). Evaluate each term: \( \frac{\partial}{\partial x}(y^3) = 0 \), \( \frac{\partial}{\partial y}(x^3) = 0 \), and \( \frac{\partial}{\partial z}(z^3) = 3z^2 \). Thus, \( abla \cdot \mathbf{F} = 3z^2 \).

Step 4: Define the Region of Integration

The region \( D \) is a cone-like structure intersecting the plane \( z=0 \) and capped by the spherical surface \( z=\sqrt{4-x^2-y^2} \). In cylindrical coordinates \( x=r\cos(\theta), y=r\sin(\theta), z=z \); where \( 0 \leq r \leq \sqrt{3} \), \( 0 \leq \theta < 2\pi \), and \( 0 \leq z \leq \sqrt{4-r^2} \).

Step 5: Convert to Cylindrical Coordinates

Substitute \( x, y, z \) in terms of \( r, \theta, z \). The differential volume element \( dV = r \, dr \, d\theta \, dz \).

Step 6: Setup and Evaluate the Triple Integral

Substitute the divergence \( 3z^2 \) into the integral: \[ \iiint_{D} 3z^2 \, dV = \int_{0}^{2\pi} \int_{0}^{\sqrt{3}} \int_{0}^{\sqrt{4-r^2}} 3z^2 \, r \, dz \, dr \, d\theta. \] Evaluate the innermost integral: \[ \int_{0}^{\sqrt{4-r^2}} 3z^2 \, dz = \left[z^3\right]_{0}^{\sqrt{4-r^2}} = (4-r^2)^{3/2}. \] Thus, the integral becomes \[ \int_{0}^{2\pi} \int_{0}^{\sqrt{3}} (4-r^2)^{3/2} \, r \, dr \, d\theta. \]

Step 7: Simplify and Evaluate Remaining Integral

Simplify and solve the remaining integrals: \[ \int_{0}^{2\pi} d\theta \int_{0}^{\sqrt{3}} (4-r^2)^{3/2} r \, dr. \] The outer integral over \( \theta \) evaluates to \( 2\pi \). Focus on: \( \int_{0}^{\sqrt{3}} (4-r^2)^{3/2} \, r \, dr \). This computation typically involves a trigonometric substitution or use of a standard integral calculus technique. Compute this integral to find the result.

Key Concepts

Vector Field

A vector field is a mathematical construct where each point in space is assigned a vector. In simpler terms, imagine a field of arrows where each arrow points in a different direction and has a different length depending on where it is in the field. These vectors represent quantities that have both magnitude and direction. For instance, wind speed across a area can be described by a vector field as each spot might experience different intensity and direction of wind.
The vector field provided in this problem is \(\mathbf{F} = y^3 \mathbf{i} + x^3 \mathbf{j} + z^3 \mathbf{k}\). Here, each component of the vector is a function of its respective coordinate \(x, y, \) or \(z\). This function of the coordinates indicates that the vector can change direction and size depending on where you examine its place in the space.

• \(y^3 \mathbf{i}\): This represents the vector component in the \(x\)-direction.
• \(x^3 \mathbf{j}\): This represents the vector component in the \(y\)-direction.
• \(z^3 \mathbf{k}\): This represents the vector component in the \(z\)-direction.

Understanding vector fields is essential because they provide insight into phenomena governed by direction and magnitude, which are vital in physics and engineering.

Outward Flux

Outward flux refers to the quantity of the vector field that flows through a surface being regarded as positive if it is directed outwards. Imagine water passing through a straight net, the water that moves out of the net represents the outward flux. Mathematically, to find the outward flux of a vector field \(\mathbf{F}\) through a surface \(S\), we use the surface integral: \[iint_{S} \mathbf{F} \cdot \mathbf{n} \; dS.\]
Here, \(\mathbf{n}\) represents the unit normal vector pointing outwards from the surface. The dot product \(\mathbf{F} \cdot \mathbf{n}\) measures the component of \(\mathbf{F}\) in the direction of \(\mathbf{n}\). The divergence theorem simplifies this surface integral to a triple integral over the volume of the region enclosed by the surface, thus converting the problem into one of volume.

• For enclosed surfaces like spheres or cylinders, we typically look for the total outward flux to determine properties like charge or mass flow.
• In the given problem, we are trying to find the flux of \(\mathbf{F}\) out of the bounding surfaces.

Understanding outward flux helps in analyzing how much of a quantity (e.g., electric field lines, fluid flow) exits from a given surface.

Cylindrical Coordinates

Cylindrical coordinates are a coordinate system that replaces the \(x\), \(y\), and \(z\) of typical Cartesian coordinates with \(r\), \(\theta\), and \(z\), respectively. This system is beneficial when dealing with problems that have cylindrical symmetry, such as those involving circles or tubes.
In cylindrical coordinates:

• \(r\): The radial distance from the \(z\)-axis.
• \(\theta\): The angle in the \(xy\)-plane from the positive \(x\)-axis.
• \(z\): The same vertical coordinate as in Cartesian coordinates.

To convert Cartesian coordinates \((x, y, z)\) into cylindrical coordinates, use:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)
• \(z = z\)

This conversion is helpful when encountering geometrical shapes like cylinders or cones. In the exercise, the bounding surface \(x^2 + y^2 = 3\) makes cylindrical coordinates more convenient because the radial symmetry mirrors cylindrical geometry.

Triple Integral

The concept of a triple integral extends the idea of integration to three dimensions and is used for finding volume or computing the total of a quantity throughout a 3D region. A triple integral is expressed generally as \(\iiint_{D} f(x, y, z) \, dV\), where \(D\) is a three-dimensional region in space.
In this task, the triple integral is applied to the divergence \(abla \cdot \mathbf{F}\), which simplifies the computation of the total outward flux from the surface enclosing the volume \(D\).

• The differential volume element \(dV\) in cylindrical coordinates is expressed as \(r \, dr \, d\theta \, dz\).
• A triple integral involves integrating successively over each variable, often starting from inside out (\(z\), then \(r\), and finally \(\theta\)).

The integral bounds are typically deduced from the geometrical boundaries of the region \(D\).
The problem involves evaluating the triple integral to yield the outward flux through the defined surface, a key demonstration of the power of the divergence theorem.