1. Separate P and Q: $\mathbf{F} = \langle P, Q \rangle$.
2. Partial Sum: Calculate $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$.
3. Check the Sign: Positive result means a Source. Negative means a Sink.
4. Result is Scalar: Your final answer must be a number or a scalar expression (No brackets!).
Find $\text{div } \mathbf{F}$ for each vector field.
$\mathbf{F} = \langle 3x^2, 5y^2 \rangle$
$\mathbf{F} = \langle xy, x+y \rangle$
$\mathbf{F} = \langle \sin x, \cos y \rangle$
For the field $\mathbf{F} = \langle x^2, y^2 \rangle$, evaluate the divergence at the given points.
At the point $(1, 1)$.
At the point $(-2, -2)$.
Calculate the divergence of the rotational field $\mathbf{F} = \langle -y, x \rangle$. Why is the result zero? Explain why a "Whirlwind" is neither a source nor a sink of energy.
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The Incompressible Air: Find the value of $k$ that makes the field $\mathbf{F} = \langle 2x, ky \rangle$ solenoidal (divergence = 0).
The gravitational field of a planet is $\mathbf{G} = \langle \frac{-x}{(x^2+y^2)^{3/2}}, \frac{-y}{(x^2+y^2)^{3/2}} \rangle$.
Task: Calculate the divergence of $\mathbf{G}$ at any point EXCEPT $(0,0)$.
(Hint: It should be zero! Gravity doesn't 'create' or 'destroy' space; it just bends it toward the center).
Objective: Explain Divergence to a younger student using a hose and a sprinkler.
The Activity:
1. Turn on the hose. Point it at a spot on the grass. "Water is flowing out ($Source$)."
2. Point to the drain in the driveway. "Water is flowing in ($Sink$)."
The Lesson: "Our life can be a hose that waters others, or a drain that takes everything for itself. God wants us to be the hose!"
Response: __________________________________________________________