Problem 24 - Thomas Calculus

Question

Compute the net outward flux of the vector field \(\mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) /\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}\) across the ellipsoid \(9 x^{2}+4 y^{2}+6 z^{2}=36\)

Step-by-Step Solution

Answer

The net outward flux of \( \mathbf{F} \) across the ellipsoid is zero.

Step 1: Identify the Problem

We are given a vector field \( \mathbf{F} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{(x^2 + y^2 + z^2)^{3/2}} \) and an ellipsoid \( 9x^2 + 4y^2 + 6z^2 = 36 \). We need to compute the net outward flux of \( \mathbf{F} \) across the surface of the ellipsoid.

Step 2: Use the Divergence Theorem

The divergence theorem allows the computation of the net flux across a closed surface \( S \) by transforming the surface integral into a volume integral: \[ \oint_{S} \mathbf{F} \cdot d\mathbf{A} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV \]. Thus, we need to first determine \( abla \cdot \mathbf{F} \).

Step 3: Compute the Divergence of F

Evaluate \( abla \cdot \mathbf{F} \) for \( \mathbf{F} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{(x^2 + y^2 + z^2)^{3/2}} \): \[ abla \cdot \mathbf{F} = abla \cdot \left(\frac{x, y, z}{(x^2 + y^2 + z^2)^{3/2}}\right) = 0 \]. This is calculated using the divergence operator on \( \mathbf{F} \), revealing that \( abla \cdot \mathbf{F} = 0 \).

Step 4: Evaluate the Volume Integral

Since \( abla \cdot \mathbf{F} = 0 \), the volume integral over any region \( V \) of \( abla \cdot \mathbf{F} \) is zero: \[ \iiint_{V} (abla \cdot \mathbf{F}) \, dV = 0 \]. Thus, indicating that the net outward flux of \( \mathbf{F} \) across the ellipsoid is zero.

Key Concepts

Vector Field

A vector field is a mathematical construct where each point in a space is associated with a vector. It gives us an idea of how a vector quantity, such as velocity, force, or electromagnetic field, is distributed throughout a region. In the problem statement, the vector field given is \( \mathbf{F} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{(x^2 + y^2 + z^2)^{3/2}} \).
This particular vector field is defined in three dimensions, using the unit vectors \(\mathbf{i}, \mathbf{j}, \text{and} \mathbf{k}\). These unit vectors represent the x, y, and z directions, respectively. The numerator \(x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\) represents a vector pointing outward from the origin, and the whole expression is then scaled by the denominator \((x^2 + y^2 + z^2)^{3/2}\), which ensures that the field diminishes as it moves away from the origin.

• Unit vectors provide direction in a vector field.
• The magnitude \((x^2 + y^2 + z^2)^{3/2}\) influences the field's strength.
• Such vector fields often occur in contexts like physics and fluid dynamics.

Flux Calculation

Flux calculation involves measuring how much of a vector field passes through a surface. In this situation, we employ the Divergence Theorem, which relates the flux through a closed surface to a volume integral over the region enclosed by the surface. To find the net outward flux of the vector field \( \mathbf{F} \) across the ellipsoid, we initially calculate the divergence of \( \mathbf{F} \). In simpler terms, divergence of a vector field estimates how much the field spreads out from a point. For our vector field, the divergence is:\[abla \cdot \mathbf{F} = 0\]When the divergence is zero, it signifies no net flow out of any small region, meaning no source or sink is present within the region. Therefore, the volume integral over any volume \( V \) becomes zero:\[\iiint_{V} abla \cdot \mathbf{F} \, dV = 0\] As a result, the computation confirms that the net flux through the ellipsoid's surface is zero.

• Divergence measures the 'spreading out' of a vector field.
• If divergence is zero, the net flux exiting any closed surface is zero.
• The Divergence Theorem simplifies flux calculations by converting them into volume integrals.

Ellipsoid

An ellipsoid is a three-dimensional geometric shape, resembling a stretched or squashed sphere. It can be described by an equation of the form \( ax^2 + by^2 + cz^2 = d \), where \(a, b, c,\) and \(d\) are constants that determine the ellipsoid's size and orientation.In this exercise, the ellipsoid is given by:\[9x^2 + 4y^2 + 6z^2 = 36\]This equation indicates our ellipsoid's dimensions and orientations. Let's break it down:

• The coefficient 9 associated with \(x^2\) means that along the x-axis, the ellipsoid has an extent or radius scaled by \(\sqrt{36/9} = 2\).
• The coefficient 4 associated with \(y^2\) indicates the dimensional stretch along the y-axis, giving a radius of \(\sqrt{36/4} = 3\).
• Lastly, the coefficient 6 associated with \(z^2\) provides the scale along the z-axis, resulting in a radius of \(\sqrt{36/6} = \sqrt{6}\).

Each unique coefficient sculpts how the ellipsoid stretches along each axis, thus shaping its overall appearance in space.