Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Notes

Parametric equations

For $x = x(t)$ and $y = y(t)$:

• Slope: $\displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ (where $\displaystyle \frac{dx}{dt} \ne 0$).
• Second derivative: $\displaystyle \frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{dx/dt}$.
• Arc length from $t = a$ to $b$: $\displaystyle \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$.

Vector-valued functions and motion in the plane

Position $\vec r(t) = \langle x(t), y(t) \rangle$.

• Velocity $\vec v(t) = \langle x'(t), y'(t) \rangle$; acceleration $\vec a(t) = \langle x''(t), y''(t) \rangle$.
• Speed $\displaystyle = |\vec v(t)| = \sqrt{(x')^2 + (y')^2}$.
• Distance traveled on $[a, b]$: $\displaystyle \int_a^b |\vec v(t)|\,dt$.
• Position from velocity: $\displaystyle x(b) = x(a) + \int_a^b x'(t)\,dt$, and the same for $y$.

Polar curves

For $r = f(\theta)$, with $x = r\cos\theta$, $y = r\sin\theta$:

• Slope: $\displaystyle \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$.
• Area inside the curve from $\alpha$ to $\beta$: $\displaystyle \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$.
• Area between two polar curves: $\displaystyle \frac{1}{2}\int_\alpha^\beta \left(r_{outer}^2 - r_{inner}^2\right)d\theta$.
• $\displaystyle \frac{dr}{d\theta} > 0$ with $r > 0$ means the curve is moving away from the origin.