Problem 4 - 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 \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=4 x \mathbf{i}+y \mathbf{j}+4 z \mathbf{k} ; D \text { the region bounded by the sphere }\\\ &x^{2}+y^{2}+z^{2}=4 \end{aligned} $$

Step-by-Step Solution

Answer

The outward flux through the sphere is \( 96 \pi \).

Step 1: Understand the Problem

We need to find the outward flux of the vector field \(\mathbf{F}=4x \mathbf{i}+y \mathbf{j}+4z \mathbf{k}\) through the surface \(S\) bounding the region \(D\), which is a sphere with equation \(x^{2}+y^{2}+z^{2}=4\). The divergence theorem is suitable for this type of problem.

Step 2: Recall the Divergence Theorem

The divergence theorem states that \( \iint_{S}(\mathbf{F} \cdot \mathbf{n}) \, dS = \iiint_{D}(abla \cdot \mathbf{F}) \, dV \), where \( \mathbf{n} \) is the outward normal vector to the surface \( S \). It allows us to transform a surface integral into a volume integral.

Step 3: Compute the Divergence of F

Calculate \( abla \cdot \mathbf{F} \). For \( \mathbf{F}=4x\mathbf{i}+y\mathbf{j}+4z\mathbf{k} \), the divergence is given by:\[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(4x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(4z) = 4 + 1 + 4 = 9. \]

Step 4: Set Up the Volume Integral

Since \( abla \cdot \mathbf{F} = 9 \) is constant, the integral simplifies to:\[ \iiint_{D} (abla \cdot \mathbf{F}) \, dV = 9 \iiint_{D} \, dV, \] where \( D \) is the volume of the sphere \( x^2 + y^2 + z^2 \leq 4 \).

Step 5: Calculate the Volume of the Sphere

The sphere \( x^2 + y^2 + z^2 = 4 \) has radius \( r = 2 \). The volume \( V \) of a sphere is \( \frac{4}{3}\pi r^3 \). For \( r = 2 \), the volume is:\[ V = \frac{4}{3} \pi (2)^3 = \frac{32}{3} \pi. \]

Step 6: Evaluate the Volume Integral

Substitute the volume of the sphere back into the integral:\[ 9 \iiint_{D} \, dV = 9 \times \frac{32}{3} \pi = 96 \pi. \] This gives us the outward flux through the surface.

Key Concepts

Vector Field

A vector field is a fundamental concept in multivariable calculus and physics, representing a spatial distribution where each point in a region of space has a vector assigned to it. In our exercise, we are dealing with the vector field \( \mathbf{F} = 4x \mathbf{i} + y \mathbf{j} + 4z \mathbf{k} \). This specific field suggests that the vectors in the field vary based on the coordinates \(x\), \(y\), and \(z\).

Key elements of vector field analysis include:

• Identifying how vectors change across the field.
• Understanding the directional nature and magnitude at various points.
• Evaluating interactions such as flux or divergence, as required in our problem.

It's important to visualize vector fields to understand how they influence objects within them, such as knowing which areas have stronger or weaker vector influences.

Surface Integral

A surface integral extends the idea of an integral from lines to surfaces. It's a critical concept for calculating quantities like flux, where you measure the "flow" of a vector field across a surface. In this exercise, we are interested in the outward flux of the vector field \( \mathbf{F} \) through a surface bounding a specific region.

The process involves:

• Determining the surface \( S \) suitable for integration, which in our scenario is a sphere.
• Calculating the integral \( \iint_{S} (\mathbf{F} \cdot \mathbf{n}) \, dS \), where \(\mathbf{n}\) is the unit normal vector pointing outwards.
• Applying the Divergence Theorem converts this surface integral problem into a volume integral, simplifying the mathematical workload.

The surface integral helps quantify how much of the vector field moves across the surface, which is especially useful in physics and engineering.

Volume Integral

The volume integral is a technique for integrating over a 3D region, allowing us to sum a field's effect over the entire volume. In our scenario, we employ the Divergence Theorem, which relates surface integrals to volume integrals.

Here's how it is applied:

• First, compute the divergence of the vector field, \( abla \cdot \mathbf{F} \), which gives us a scalar field that represents the rate of expansion of the vector field.
• The integral \( \iiint_{D} (abla \cdot \mathbf{F}) \, dV \) is then evaluated over the volume \( D \), in our case, the solid sphere.
• The volume integral \( 9 \iiint_{D} \, dV \) accounts for the entire enclosed space, translating surface calculations into a simpler volume computation.

The outcome of this integral gives us the total flux exiting the closed surface, providing an efficient way to work across geometrically complex boundaries.

Sphere

The sphere in this problem acts as the boundary \( S \), defined by \( x^2 + y^2 + z^2 = 4 \). This equation describes a sphere centered at the origin with a radius of 2 units. Understanding the geometry of the sphere is crucial because:

• The symmetry of the sphere simplifies calculations, particularly when applying the Divergence Theorem.
• The radius directly influences the sphere's volume, \( \frac{4}{3}\pi r^3 \), which was determined to be \( \frac{32}{3}\pi \) for a radius of 2.

Calculating the volume of our spherical region is vital. It plugs directly into the integral for computing the total flux. Spheres frequently appear in physics for modeling situations with isotropic properties, such as gravitational fields or radiation patterns.