Vector-Valued Functions: An Introduction to Multi-Dimensional Calculus

Week 2: Vector-Valued Functions

Goals for this Unit

• Define vector-valued functions (also known as parametric curves).

• Sketch vector-valued functions in two or three dimensions.

• Determine if two vector-valued functions collide (same position at same time) or intersect (same position at possibly different times).

The Calculus of Moving Objects

Describing Multi-Dimensional Motion

• Problem: How to describe an object moving in more than one dimension?

• Single-variable limitation: Previously, we explored position $x(t)$ changing with time $t$, leading to velocity $x'(t)$ or $v(t)$.

  • This approach is too restrictive for physics problems involving motion in 2 or more dimensions.

• Need: A consistent method to describe changing positions, velocities, and accelerations in multi-dimensional space.

Introductory Problem (Three-Dimensional Motion)

• Consider an object moving in 3D space with coordinates at time $t ag{t hinspace extgreater hinspace= hinspace 0}$ given by the vector function $hinspace hinspace r(t) = hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspace hinspacect hinspace hinspace hinspacect hinspace hinspace hinspace hinspace hinspacect hinspace hinspacect hinspace hinspacect hinspace hinspace hinspace hinspacect hinspace hinspace hinspacect hinspace hinspacect hinspace hinspacect hinspace hinspacect hinspace hinspacet, hinspace hinspace hinspaced, hinspace e hinspace^t hinspace hinspace angle$. This problem will be revisited at the end of the unit. Key questions arising from this scenario:

  • What would the graph of its trajectory look like?

  • What is its speed at, for instance, time $t = 0$?

  • If this object is a vehicle, when would a passenger experience the greatest acceleration?

  • If a passenger were to fall out at $t = 1$ in a zero-gravity environment, what would happen to them?

Four Ways to Represent a Function

Definition of a Function

• A function is a rule or process that assigns a corresponding output to each input.

• Functions can take various forms.

Example: Height of a Green Dot on a Wheel

• Scenario: A wheel of radius $1$, centered at the origin, with a green dot G at its extreme right.

• Rotation: The wheel turns counter-clockwise by an angle $heta$.

• Function: The height $h( heta)$ of the dot above (or below) the x-axis varies with the angle $heta$.

Methods of Function Representation

• From the example, there are at least four ways to represent a function:

  • Verbal description: As stated in a problem.

  • Table: A list of input-output pairs.

  • Graph: A visual representation on a coordinate plane.

  • Formula: An algebraic rule.

• Crucially, a function is not necessarily defined by a formula alone.

Characteristics of Functions

• In all representations, a function involves two varying quantities:

  • Input: The independent variable, which