Chapter 12: Vector-Valued Functions and Motion in Space

Section 12.1: Curves in Space and Their Tangents

This section introduces vector-valued functions to describe curves in three-dimensional space, focusing on their properties, limits, continuity, and tangents.

• Position Vector: The position of a particle in space at time t is given by the vector function:

• Space Curves: A space curve is traced by the tip of the position vector as t varies. Examples include helical, parabolic, and toroidal curves, each defined by specific component functions.

• Limit of a Vector Function: The limit of as is if for every , there exists such that whenever .

• Continuity: is continuous at if .

• Tangent Vector: The tangent vector to the curve at is the derivative of the position vector: or, component-wise:

Velocity, Speed, and Acceleration

These concepts generalize motion in space using derivatives of the position vector.

• Velocity Vector: (tangent to the curve, direction of motion)

• Speed: The magnitude of velocity:

• Acceleration Vector:

• Unit Tangent Vector: (direction of motion at time t)

Differentiation Rules for Vector Functions

Vector functions follow rules analogous to scalar functions, with additional rules for dot and cross products.

Special Properties

• If a particle moves on a sphere and is a differentiable function of time, then (the position and velocity vectors are orthogonal).

• If is a differentiable vector function of constant length, then .

Section 12.2: Integrals of Vector Functions; Projectile Motion

This section covers integration of vector functions and applies these concepts to projectile motion.

• Indefinite Integral: The set of all antiderivatives of :

• Component-wise Integration: Integrate each component separately:

• Definite Integral: Over :

Projectile Motion

• Initial Position and Velocity: At ,

• Position as a Function of Time:

• Maximum Height:

• Flight Time:

• Range:

Section 12.3: Arc Length in Space

This section introduces the concept of arc length for curves in space, providing formulas for calculation.

• Arc Length of a Smooth Curve: For , :

• Alternative Arc Length Formula:

• Arc Length Parameter: The directed distance along the curve from to is:

Example: Calculating Arc Length

Given for , the arc length is:

• Compute

• Find

• Integrate:

Additional info: These notes are based on textbook slides and provide a concise summary of vector-valued functions, their calculus, and applications to motion in space, suitable for college-level Calculus students.