Problem 18 - Thomas Calculus

Question

In Exercises 9- \(20,\) use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Thick sphere \(\quad \mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) / \sqrt{x^{2}+y^{2}+z^{2}}\) \(D :\) The region \(1 \leq x^{2}+y^{2}+z^{2} \leq 4\)

Step-by-Step Solution

Answer

The outward flux is zero.

Step 1: Understand the Divergence Theorem

The Divergence Theorem relates the flux of a vector field across a closed surface to the volume integral of the divergence of the field inside the surface. Mathematically, it states: \( \iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV \), where \( V \) is the volume inside surface \( S \) and \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \).

Step 2: Compute the Divergence

First, find the divergence of the given vector field \( \mathbf{F} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{\sqrt{x^2 + y^2 + z^2}} \). The divergence \( abla \cdot \mathbf{F} \) is computed as \( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \). Given \( F_x = \frac{x}{\sqrt{x^2+y^2+z^2}} \), compute its derivative and likewise for \( F_y \) and \( F_z \).

Step 3: Simplify the Expression

Compute the partial derivatives of each component. For example, \( \frac{\partial}{\partial x}\left(\frac{x}{\sqrt{x^2+y^2+z^2}}\right) = \frac{y^2 + z^2}{(x^2+y^2+z^2)^{3/2}} \). Similarly, calculate for \( y \) and \( z \). After simplification, each term simplifies, and the total divergence is zero: \( abla \cdot \mathbf{F} = 0 \).

Step 4: Apply the Divergence Theorem

Since the divergence \( abla \cdot \mathbf{F} \) is zero, according to the Divergence Theorem, the volume integral is \( \iiint_{V} 0 \, dV = 0 \). This implies that the outward flux of \( \mathbf{F} \) across the boundary of region \( D \) is zero.

Key Concepts

Vector Field

A vector field is essentially a collection of vectors associated with each point in a space. Think of it like a breeze of wind where every point in the atmosphere has a specific direction and magnitude. In mathematics, a vector field can be represented using unit vectors like \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) that denote the x, y, and z directions, respectively.

In the given exercise, the vector field \( \mathbf{F} \) is expressed as \( F= \frac{x \mathbf{i}+y \mathbf{j}+z \mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}} \). This equation describes vectors that emanate outwards and have a unit length at each point in space.

Understanding vector fields is essential as they are used to model various physical phenomena like gravitational fields, fluid flow, and electromagnetic fields. The behavior and properties of a vector field determine how it interacts within its defined space.

Flux

Flux measures the flow of a vector field through a surface. Imagine water flowing through a net. The amount of water passing through the net represents the flux. In mathematics, flux is calculated by integrating the dot product of the vector field \( \mathbf{F} \) with a surface element \( d\mathbf{S} \) across the surface.

More formally, the flux of \( \mathbf{F} \) across a closed surface \( S \) is given by \( \iint_{S} \mathbf{F} \cdot d\mathbf{S} \).

• If the flux is positive, more field lines are exiting the surface.
• If negative, more lines are entering than leaving.
• A zero flux implies no net flow through the surface.

The concept of flux is crucial in physical sciences where we study energy, heat, and electricity flowing from one point to another.

Divergence

Divergence measures how a vector field diverges, or spreads out, from a point. It provides an understanding of the tendency of fluids to either converge towards or diverge away from certain points within a region.

For a given vector field \( \mathbf{F} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{\sqrt{x^2 + y^2 + z^2}} \), divergence is expressed as \( abla \cdot \mathbf{F} \). By computing \( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \), we gain valuable insights.

• If \( abla \cdot \mathbf{F} > 0 \), the field is diverging more from the point.
• If \( abla \cdot \mathbf{F} < 0 \), the field is converging.
• For zero divergence, the field has no net flux.

In the exercise, the divergence was zero, indicating that the vector field does not stretch or compress at any point within the region.

Volume Integral

A volume integral extends the concept of integration to three dimensions to evaluate a function over a volume. This becomes especially useful when dealing with volume quantities in vector fields, such as in applying the Divergence Theorem.

The exercise uses a volume integral to compute the overall behavior of the divergence within the region \( D \). It turns the three-dimensional space into a manageable form used for calculations.

Mathematically, it corresponds to the integral \( \iiint_{V} abla \cdot \mathbf{F} \, dV \).

• It involves slicing the volume into tiny parts and summing the effects over them.
• If the divergence is zero, as in this exercise, the integral will naturally conclude with zero, indicating no net "spreading out" of the vector field in the space.

The volume integral, when paired with the Divergence Theorem, provides a powerful tool to calculate and analyze the characteristics of vector fields across complex volumes.