Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Parametric equation

A coordinate representation where x and y are given separately as functions of a parameter: x=f(t), y=g(t).

Parameter (t)

The independent variable in a parametric description; it controls which point (x(t),y(t)) is on the curve (often interpreted as time).

Parametric curve

The set of points (x(t),y(t)) traced as t varies over a specified interval.

Dependent vs. independent variables (parametric)

In parametrics, x and y depend on t; t is independent, while x(t) and y(t) are dependent variables.

Orientation (direction of tracing)

The direction the curve is traced as the parameter increases (e.g., clockwise vs. counterclockwise).

Reparameterization

A different parametric description of the same geometric curve that can change tracing speed and/or orientation without changing the set of points (if the interval covers the same points).

Looping / retracing

When different parameter values produce the same point (x(t),y(t)), causing the curve to cross itself or repeat portions.

Eliminating the parameter

Solving for t from one parametric equation and substituting into the other to get a rectangular relationship between x and y.

Rectangular equation (from parametrics)

An equation relating x and y directly (no parameter), obtained by eliminating t.

Information lost when eliminating the parameter

Elimination typically removes orientation and can hide which portion of the curve is traced or whether the curve is retraced.

Parameter-interval restriction

The chosen t-interval (or θ-interval) determines which points are included; changing the interval can trace only part of the same rectangular curve.

Slope of a parametric curve (tangent slope)

The derivative dy/dx computed by the chain rule: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Vertical tangent (parametric)

Occurs when dx/dt = 0 and dy/dt ≠ 0 (slope is undefined/vertical).

Horizontal tangent (parametric)

Occurs when dy/dt = 0 and dx/dt ≠ 0 (slope equals 0).

Common slope mistake (dy/dt)

dy/dt is NOT the slope of the curve in the xy-plane; slope requires dy/dx = (dy/dt)/(dx/dt).

Second derivative for parametric curves

d²y/dx² = (d/dt(dy/dx)) / (dx/dt). Differentiate dy/dx with respect to t, then divide by dx/dt.

Concavity (parametric)

Determined by the sign of d²y/dx²: positive means concave up (locally as y vs. x), negative means concave down.

Signed area under a parametric curve

Area accumulated using A = ∫ y dx; for parametrics, compute using the parameter to account for how x changes.

Signed area formula (parameter form)

A = ∫_a^b y(t)·(dx/dt) dt. The sign depends on the direction the curve moves in x.

Direction effect on signed area

If dx/dt < 0 (moving right-to-left), the integral ∫ y(t) x'(t) dt contributes negative (signed) area.

Arc length (parametric)

Distance along a parametric curve: L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt.

Speed integrand for arc length

For planar parametrics, the quantity √((dx/dt)² + (dy/dt)²) is the speed along the curve (magnitude of velocity).

Common arc length algebra error

In arc length/speed, you square the derivatives (dx/dt and dy/dt), add, then take a square root; do not square x or y themselves.

Velocity components (2D motion)

If x=x(t), y=y(t), then vx = dx/dt and vy = dy/dt (components of the velocity vector).

Speed (magnitude of velocity)

speed = √((dx/dt)² + (dy/dt)²); a scalar measuring how fast the particle moves along the path.

Acceleration components (2D motion)

ax = d²x/dt² and ay = d²y/dt² (components of the acceleration vector).

At rest / “stops” condition

A particle stops only when the velocity vector is zero, meaning all velocity components are 0 simultaneously.

Polar coordinates

A coordinate system using (r, θ), where r is directed distance from the origin (pole) and θ is the angle from the positive x-axis.

Pole and polar axis

The pole is the origin in polar coordinates; the polar axis is the ray θ=0 (the positive x-axis direction).

Equivalent polar representations

The same point can be written with angles differing by 2πk, and also by using negative r (which flips the point through the pole).

Polar–rectangular conversion (with quadrant awareness)

x=r cosθ, y=r sinθ; also r²=x²+y² and tanθ=y/x, but θ must be chosen to match the correct quadrant.

Polar curve (r = f(θ))

A curve described by giving radius as a function of angle; points are plotted at angle θ and distance r from the pole.

Polar symmetry tests

Quick checks: symmetry about polar axis if θ→−θ leaves equation unchanged; about θ=π/2 if θ→π−θ unchanged; about pole if r→−r (or θ→θ+π) unchanged.

Polar circle example: r = 2cosθ

Converts to x²+y²=2x, i.e., (x−1)²+y²=1 (a circle centered at (1,0) with radius 1).

Polar as a parametric curve

Treat θ as the parameter: x(θ)=r(θ)cosθ and y(θ)=r(θ)sinθ, so parametric calculus applies.

Polar derivatives dx/dθ and dy/dθ

If r=r(θ): dx/dθ = r'(θ)cosθ − r(θ)sinθ, and dy/dθ = r'(θ)sinθ + r(θ)cosθ.

Slope of a polar curve

dy/dx = (dy/dθ)/(dx/dθ) = (r' sinθ + r cosθ)/(r' cosθ − r sinθ).

Polar area formula

Area enclosed by a polar curve: A = (1/2)∫_α^β (r(θ))² dθ.

Area between two polar curves

A = (1/2)∫α^β (router(θ)² − r_inner(θ)²) dθ (outer minus inner).

Intersection angles / bounds in polar area

To find bounds for regions between curves, solve r1(θ)=r2(θ) and confirm the interval matches the intended region on the sketch.

Full trace / retracing issue (polar)

A “full trace” may occur on an interval smaller than 0 to 2π (or the curve may retrace), so choosing correct bounds is essential.

Polar arc length formula

Arc length of r=r(θ): L = ∫_α^β √(r(θ)² + (dr/dθ)²) dθ.

Vector-valued function

A function whose output is a vector, e.g., r(t)=⟨x(t),y(t),z(t)⟩, commonly used to represent position.

Position, velocity, and acceleration vectors

Position r(t); velocity v(t)=r'(t); acceleration a(t)=v'(t)=r''(t).

Component-wise calculus for vectors

Differentiate/integrate vector functions by components: r'(t)=⟨x',y',z'⟩ and ∫r(t)dt=⟨∫x,∫y,∫z⟩+C (C is a constant vector).

Displacement vs. total distance traveled

Displacement is r(b)−r(a) (a vector); total distance traveled is arc length: ∫_a^b |r'(t)| dt (a scalar).

Unit tangent vector

T(t)=r'(t)/|r'(t)|; a length-1 vector pointing in the direction of motion along the curve.

Curvature (how sharply a path turns)

Measures turning rate of the path: for plane parametrics κ(t)=|x'y''−y'x''|/((x')²+(y')²)^(3/2); in 3D κ(t)=|r'×r''|/|r'|³.

Tangential acceleration (a_T)

The rate of change of speed: a_T = d/dt(|v(t)|). It measures how the motion speeds up or slows down.

Normal acceleration (a_N)

The turning component of acceleration: a_N = κ|v(t)|². It can be nonzero even when speed is constant (e.g., circular motion).