1. Parameterize the Path ($C$): Write $x(t)$ and $y(t)$ for the journey.
2. Calculate Velocity ($d\mathbf{r}$): Find $\mathbf{r}'(t) = \langle dx/dt, dy/dt \rangle$.
3. Substitute: Plug your $x(t)$ and $y(t)$ into the Vector Field $\mathbf{F}$.
4. Dot Product & Integrate: Sum the agreement $\mathbf{F} \cdot \mathbf{r}'(t)$ over the time interval.
Find the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for each scenario.
The Straight Path: Field $\mathbf{F} = \langle 3, 2 \rangle$. Path $\mathbf{r}(t) = \langle t, 2t \rangle$ from $t=0$ to $t=1$.
The Rising Wind: Field $\mathbf{F} = \langle y, x \rangle$. Path is a line from $(0,0)$ to $(1,1)$.
Hint: $\mathbf{r}(t) = \langle t, t \rangle$ for $t \in [0, 1]$.
Calculate the circulation around the unit circle ($x=\cos t, y=\sin t$) for each field.
The Whirlpool: $\mathbf{F} = \langle -y, x \rangle$ from $t=0$ to $2\pi$.
The Radial Push: $\mathbf{F} = \langle x, y \rangle$ from $t=0$ to $2\pi$.
In the "Radial Push" problem, your answer should be 0. Try to visualize the arrows pointing out from the center while you walk in a circle around it. Why is the agreement zero? Does God ever push us sideways relative to His call?
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Find the circulation of $\mathbf{F} = \langle y, -x \rangle$ around the ellipse $\mathbf{r}(t) = \langle 2\cos t, 3\sin t \rangle$ for one full loop.
Objective: Explain Circulation to a younger student using a paper windmill or pinwheel.
The Activity:
1. Blow on the side of the pinwheel. It spins. "This is positive circulation."
2. Blow directly at the center of the pinwheel. It doesn't spin. "This is zero circulation."
The Lesson: "Worship is like the wind hitting the side of our heart to make it spin. We have to catch the wind at the right angle to get the energy!"
Response: ___________________________________________________________