Vector-Valued Functions and Calculus

Flashcards for reviewing vector-valued functions and calculus concepts.

Parametric Curve Orientation

The direction in which a parametric curve is traced as the parameter t increases.

Circular Helix

A corkscrew-shaped curve that wraps around a right circular cylinder, often described by the parametric equations x = a cost, y = a sin t, z = ct.

Torus Knot

A parametric curve in 3-space whose depiction may require a tube plot to avoid visual ambiguity about intersections.

Tube Plot

A graph of a parametric curve enclosed within a thin tube, aiding visualization by clarifying spatial relationships.

Twisted Cubic

A curve defined by the parametric equations x = t, y = t^2, z = t^3.

Vector-Valued Function

A function that associates vectors with real numbers, often used to describe parametric curves; can be in 2-space or 3-space.

Component Functions

The functions x(t), y(t), and z(t) that define the components of a vector-valued function r(t).

Natural Domain (Vector-Valued Function)

The intersection of the natural domains of the component functions of a vector-valued function r(t).

Graph of a Vector-Valued Function

The parametric curve described by the component functions of the vector-valued function.

Position Vector

A vector r(t) whose initial point is at the origin and whose terminal point traces out a curve C as the parameter t varies.

Two-Point Vector Form of a Line

A vector equation for the line passing through the terminal points of vectors r0 and r1.

Line Segment Vector Form

r = (1 − t)r0 + tr1 for 0 ≤ t ≤ 1 represents the line segment in 2-space or 3-space that is traced from r0 to r1.

Limit of a Vector-Valued Function

The vector L that r(t) approaches as t approaches a, defined such that limt→a |r(t) − L| = 0.

Continuity of a Vector-Valued Function

A vector-valued function r(t) is continuous at t = a if limt→a r(t) = r(a).

Derivative of a Vector-Valued Function

r'(t) = limh→0 (r(t + h) − r(t))/h, representing a vector tangent to the curve traced by r(t).

Tangent Vector

A vector r'(t0) that, if it exists and is non-zero, is tangent to the graph of r(t) at r(t0).

Rules of Differentiation (Vector-Valued Functions)

Rules analogous to those for real-valued functions, such as linearity, constant multiple rule, sum/difference rule, and scalar multiple rule.

Derivatives of Dot and Cross Products

Formulas for differentiating dot products and cross products of vector-valued functions.

Definite Integral of a Vector-Valued Function

The integral of r(t) from a to b is a vector whose components are the definite integrals of the component functions of r(t).

Fundamental Theorem of Calculus (Vector Form)

If R(t) is an antiderivative of r(t), then ∫ab r(t) dt = R(b) − R(a).

Smooth Parametrization

A curve represented by r(t) where r'(t) is continuous and r'(t) ≠ 0 for any allowable t.

Arc Length Parametrization

A parametric representation of a curve using arc length s as the parameter.

Chain Rule (Vector-Valued Functions)

If r(t) is a vector-valued function and t = g(τ), then dr/dτ = (dr/dt)(dt/dτ).

Positive Change of Parameter

A change of parameter t = g(τ) in which dt/dτ > 0 for all τ, preserving the orientation of a parametric curve.

Negative Change of Parameter

A change of parameter t = g(τ) in which dt/dτ < 0 for all τ, reversing the orientation of a parametric curve.

Arc Length Parameter Change Formula

s = ∫tt0 |dr/du| du: a positive change of parameter from t to s, where s is an arc length parameter.

ds/dt

The length of the tangent vector: ds/dt= |dr/dt|

Unit Tangent Vector

T(t) = r'(t) / |r'(t)|, a unit vector tangent to the curve and pointing in the direction of increasing parameter.

Unit Normal Vector

N(t) = T'(t) / |T'(t)|, a unit vector normal to the curve, pointing in the direction of T'(t).

Inward Unit Normal

the unit normal vector is the one that points inward toward the concave side of the curve in 2-space

Binormal Vector

B(t) = T(t) x N(t), a unit vector orthogonal to both T(t) and N(t), creating a right-handed coordinate system.

Curvature

κ(s) = |dT/ds| = |r''(s)|, a measure of how sharply a curve bends, defined in terms of arc length parametrization.

Osculating Circle

The circle of radius ρ = 1/κ sharing a common tangent with C at P, and centered on the concave side of the curve at P

Velocity

v(t) = dr/dt, the instantaneous rate of change of position with respect to time.

Acceleration

a(t) = dv/dt = d^2r/dt^2, the instantaneous rate of change of velocity with respect to time.

Speed

|v(t)| = ds/dt, the instantaneous rate of change of arc length with respect to time.

Displacement

Δr = r(t2) − r(t1), the change in the object's position during the time interval t1 <= t <= t2

Tangential Component of Acceleration

aT = d^2s/dt^2 = v · a / |v|, measures the rate of change of speed.

Normal Component of Acceleration

aN = κ (ds/dt)^2 = |v x a| / |v|, measures the rate of change of direction.

Newton's Second Law of Motion

F = ma, used for modeling projectile motion under gravity.