Volume 4: The Dimensions of Spirit

Workbook 35.3: The Divergence Theorem

Directives for the Impartation Officer:

1. Calculate the Divergence: Find $\nabla \cdot \mathbf{F}$ first.
2. Evaluate the Volume: Find the volume of the closed surface ($E$).
3. Multiply: If Divergence is constant, $\text{Flux} = \text{div } \mathbf{F} \times \text{Volume}$.
4. Integrate: If Divergence is not constant, solve $\iiint \text{div } \mathbf{F} dV$.

Part I: The Cube Flux

Find the outward flux of $\mathbf{F} = \langle 2x, 3y, z \rangle$ across a cube defined by $0 \le x, y, z \le 1$.

Step 1: Find the divergence $\nabla \cdot \mathbf{F}$.

$\frac{\partial}{\partial x}(2x) = 2$
$\frac{\partial}{\partial y}(3y) = 3$
$\frac{\partial}{\partial z}(z) = 1$
$\text{div } \mathbf{F} = 2 + 3 + 1 = \mathbf{6}$.

Step 2: Use the Divergence Theorem to find the Total Flux.

$\text{Flux} = \iiint_{V} 6 dV$
$\text{Volume of cube} = 1 \times 1 \times 1 = 1$.
$\text{Flux} = 6 \times 1 = \mathbf{6}$.
The Integrity Check:

If you have a vector field with **Zero** divergence ($\\nabla \cdot \mathbf{F} = 0$), what is the total flux across ANY closed surface? Does it matter if the surface is a cube or a sphere? Explain what this tells you about a "Pure Conduit" life.

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Part II: The Sphere of Grace

Find the flux of $\mathbf{F} = \langle x, y, z \rangle$ across a sphere of radius $R = 2$.
Note: Volume of a sphere $= \frac{4}{3}\pi R^3$.

The Calculation:
1. Find divergence.
2. Calculate Volume.
3. Multiply.

$\text{div } \mathbf{F} = 1+1+1 = 3$.
Volume $= \frac{4}{3}\pi (2^3) = \frac{32}{3}\pi$.
Flux $= 3 \cdot \frac{32}{3}\pi = \mathbf{32\pi}$.

Part III: Identifying the Sink

If $\text{div } \mathbf{F} = -5$ everywhere inside a vessel of volume 10... what is the total flux through the skin of the vessel?

Flux $= -5 \times 10 = -50$.
Does air flow OUT or IN?

Part IV: The Challenge (The Tabernacle Presence)

The Glowing Room

The Spirit field inside a room is $\mathbf{F} = \langle x^2, y^2, z^2 \rangle$.
The room is a box: $0 \le x \le 2, 0 \le y \le 2, 0 \le z \le 2$.
Task: Find the total outward flux of the Spirit through the walls.

1. Find $\text{div } \mathbf{F} = 2x + 2y + 2z$...
2. Set up the Triple Integral: $\int_{0}^{2} \int_{0}^{2} \int_{0}^{2} (2x+2y+2z) dx dy dz$...
3. Solve...

Part V: Transmission (The Echad Extension)

Teacher Log: The Sieve Test

Objective: Explain the Divergence Theorem to a younger student using a colander and water.

The Activity:
1. Put your hands inside the colander.
2. Turn on the faucet above your hands.
3. "The water leaving the holes is exactly the same as the water hitting my hands."

The Lesson: "Our life is like this colander. What God puts in our heart is exactly what the world sees coming out of our skin."


Response: ___________________________________________________________

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