1. Separate P and Q: $\mathbf{F} = \langle P, Q \rangle$.
2. Calculate Cross-Partials: Find $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$.
3. Subtract: $\text{curl } \mathbf{F} = Q_x - P_y$.
4. Interpret the Sign: Positive result means Counter-Clockwise (CCW) spin. Negative means Clockwise.
Find the 2D Curl for each vector field.
$\mathbf{F} = \langle -y^2, x^2 \rangle$
$\mathbf{F} = \langle x^2 y, y^2 x \rangle$
$\mathbf{F} = \langle e^x, \sin y \rangle$
Calculate the curl of the gradient field $\mathbf{F} = \langle 2x, 2y \rangle$ (the gradient of $x^2+y^2$). Why is the result zero? Explain why a field that points "Straight Away" from a center cannot make a paddle-wheel spin.
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Evaluate the curl of $\mathbf{F} = \langle -y, x \rangle$ at the given points.
At the point $(10, 10)$.
At the point $(0, 0)$.
The Whirlwind: Find the value of $k$ that makes the field $\mathbf{F} = \langle ky, x \rangle$ irrotational (Curl = 0).
A 3D vector field is $\mathbf{F} = \langle -y, x, z \rangle$.
Task: Find the 3D Curl vector $\nabla \times \mathbf{F}$.
Recall: $\text{curl } \mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle$.
Objective: Explain Curl to a younger student using a bowl of cereal and a spoon.
The Activity:
1. Put a floating cereal piece in the bowl.
2. Stir the milk in a circle.
3. Watch the cereal piece spin around its own center as it travels.
The Lesson: "Worship is like the spoon. it makes the whole bowl spin so that every little piece feels the energy of the center."
Response: ___________________________________________________________