Lecture on Vector Functions in Calculus 3

Introduction to Vector Functions

• A new chapter in calculus learning, specifically Calculus 3 (Cal 3).
• Focus on understanding derivatives in 3D using vectors.
• Vector Functions: An extension of parametric equations, using vectors.
• Vector functions use parameters (often denoted as 't') to define functions.

Parametric Equations Recap

• A parametric equation involves a parameter (t) to define variables like x and y.
• Parameter (t): Provides a common domain for functions.
• X and Y in parametric equations depend on 't'.

Transitioning to 3D

• Adding another variable to move from 2D (x, y) to 3D (x, y, z).
• Notation: Often uses R^3 to denote 3D space.

Understanding Vector Functions

• Vector functions can be expressed as positions depending on 't' (i.e., R(t) = (x(t), y(t), z(t)) ).
• Each component (x, y, z) is a function of 't'.
• Vectors defined by components (i, j, k) where:
  • x(t) associated with i
  • y(t) associated with j
  • z(t) associated with k
• Vector Functions: Define space curves traced by terminal points of vectors.

Sketching Vector Functions

• Identify x, y, z components.
• Use two components to define a curve or surface.
• Use third component to sketch vector curve on the surface.
• Orientation: Determined by the parameter interval and direction of time 't'.

Examples and Exercises

• Example 1: Plotting simple vector functions.

  • Define x, y, z from parametric equations.
  • Identify domain restrictions.
  • Plot points and determine curve orientation.

• Example 2: More complex 3D vector functions.

  • Sketch on surfaces like cylinders or cones.
  • Use substitution to manage components and eliminate parameters.

Limits and Continuity with Vector Functions

• Limits: Evaluate limit of each component individually (x, y, z).
• Continuity: Identified by common domain where all component functions are continuous.

Conclusion and Key Takeaways

• Vector functions extend calculus into 3D space using parametric ideas.
• Sketching relies on understanding surfaces and plots in multi-dimensional space.
• Mastery of vector functions involves recognizing vector notation and applying calculus principles like limits and continuity.