An equation where x and y are defined in terms of a third variable, usually time (t).

A function that maps a real number to a vector in a vector space, often representing position, velocity, or acceleration.

A two-dimensional coordinate system where a point is located by its distance (r) from the origin and angle (θ) from the positive x-axis.

The distance along a curve defined by a function between two points.

A function of the form r = f(θ), where r is the distance from the origin and θ is the angle.

A two-dimensional coordinate system in which the position of a point is determined by the distance from the origin (r) and the angle (theta θ) between the positive x-axis and the line connecting the point to the origin, counterclockwise.

Time (t).

x and y.

The xy-plane, also known as ℝ^2.

Position, velocity, and acceleration of an object moving in space.

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}$

$L = \int_{a}^{b} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$

$A = \frac{1}{2} \int_{a}^{b} r^2 d\theta$

$A = \frac{1}{2} \int_{a}^{b} (R^2 - r^2) d\theta$

$x = r \cos(\theta), y = r \sin(\theta)$

$r = \sqrt{x^2 + y^2}$

$\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}$

r'(t) = <f'(t), g'(t)>

∫v(t) dt = <∫f(t) dt, ∫g(t) dt>

Parametric: x and y are functions of t | Cartesian: y is a function of x

Vector-Valued: Integrate each component separately, result is a vector | Scalar: Integrate a single function, result is a scalar

Cartesian: Integrate difference of functions wrt x, A = ∫(top - bottom) dx | Polar: Integrate difference of squared polar functions wrt θ, A = 1/2 ∫(R² - r²) dθ

Velocity: Vector quantity with magnitude and direction | Speed: Scalar quantity, magnitude of velocity

Position Vector: Represents location at a given time | Velocity Vector: Represents rate of change of position at a given time

Polar: Defined by radius and angle (r, θ) | Cartesian: Defined by horizontal and vertical distance (x, y)

Parametric: Use chain rule, $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ | Cartesian: Direct differentiation, $\frac{dy}{dx}$

Parametric: Integrate with respect to t | Cartesian: Integrate with respect to x

Vector-valued: Output is a vector | Scalar: Output is a single number

Cartesian: Integrate $\sqrt{1 + (dy/dx)^2}$ dx | Parametric: Integrate $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ dt