Problem 64 - Calculus: Early Transcendental Functions

Question

In geometric terms, determine whether the flux integral of \(\quad \mathbf{F}(x, y, z)=\langle x, y, z\rangle\) across the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) is 0

Step-by-Step Solution

Answer

The flux integral of the given vector field across the hemisphere is not 0. It equals \(\frac{\pi}{2}\)

Step 1: Calculate the Divergence of the Vector Field

The vector field \(\mathbf{F}(x, y, z)=\langle x, y, z\rangle\) has a divergence calculated by \(\nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3\)

Step 2: Apply the Divergence Theorem for the Hemisphere

By applying the divergence theorem on the hemisphere, we take the triple integral over the half sphere (with radius 1) as \(\frac{1}{2}\) times the triple integral over the total sphere of radius 1, which is \(\frac{1}{2} \times \frac{4}{3} \pi r^{3} = \frac{1}{2} \times \frac{4}{3} \pi = \frac{2}{3} \pi\). Now, the divergence over the volume enclosed by the hemisphere is \(3 \times \frac{2}{3} \pi = 2\pi\)

Step 3: Determine the Flux Integral over the Disk

We compute the integral \(\iint_{D} \mathbf{F} \cdot d\mathbf{S}\) across the disk at the bottom, where the unit normal vector is always \(n = (0, 0, -1)\). The flux across the disk is \(\iint_{D} \mathbf{F} \cdot (-\mathbf{k}) \,dA = - \iint_{D} z \,dA = - \iint_{D} \,dA = - \pi r^{2} = - \pi\)

Step 4: Sum up the Flux

|The total flux integral across the entire sphere is the sum of the flux integral over the hemisphere and the flux integral over the disk. Hence, \(2\pi - \pi = \pi\). However, since, in the beginning, we considered the whole sphere instead of the original hemisphere, the flux through the hemisphere is actually half of this value, i.e. \(\frac{\pi}{2}\)

Key Concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus, making it easier to compute flux integrals over complex surfaces. To understand this, think of the theorem as a bridge connecting surface integrals to volume integrals. It relates the flux of a vector field across a closed surface to the divergence of the vector field inside the volume.
The mathematical expression of the Divergence Theorem is:
\[\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV\]This equality shows that the total outward flux across a closed surface (\(\partial V\)) is equal to the triple integral of the divergence (\(abla \cdot \mathbf{F}\)) throughout the volume (\(V\)).
For the exercise, this theorem helps determine the flux across a hemisphere by first calculating the divergence inside the sphere's volume.

Vector Fields

Vector fields are ubiquitous in mathematics and physics, describing how vectors vary over a region of space. A vector field assigns a vector to each point in space, often visualized as arrows with direction and magnitude.
In the exercise, the vector field is given by \(\mathbf{F}(x, y, z) = \langle x, y, z\rangle\). This indicates that each point

• Has a vector where its direction points outward.
• The magnitude equals the distance from the origin.

Understanding the properties of this vector field is crucial because it affects how the field behaves on a given surface such as a hemisphere. The exercise specifically looks at how this vector field's flux changes through the hemisphere and disc.

Hemisphere

In this specific context, a hemisphere means half of a full sphere, typically cut along a plane such as \(z=0\), where the upper half is the focus.
A hemisphere has two parts:

• The curved surface (like a dome).
• A flat base (like the top of a table).

In the exercise, the hemisphere is defined by \(z = \sqrt{1 - x^2 - y^2}\), which represents the top half of a sphere with radius 1. Analyzing flux over a hemisphere contrasts with a full sphere due to its boundary (flat base) which contributes separately to flux calculations.
The distinction is crucial because it splits the work into understanding flux over the dome's surface and the base.

Flux Calculation

Flux calculation measures how much of a vector field passes through a given surface. It is a vital concept when understanding fields and their interactions with objects.
In the exercise, the flux integral computes how much of the vector field \(\mathbf{F}(x, y, z) = \langle x, y, z\rangle\) passes through the hemisphere and the circular disk at the base.
First, apply the Divergence Theorem to the hemisphere, which simplifies calculations by using the divergence inside the volume rather than the surface. Next, calculate the flux over the disk base, noting the inward-pointing normal to the disk.
The contributions from each part are summed to find the total flux, which, when accurately evaluated, reveals specific physical or mathematical insights about the field's distribution and interaction with the surface.