By the end of Week : You would learn how to solve many problems involving limits, derivatives and integrals of vector-valued functions and questions regarding parametrization of curves. These are just few examples: Find parametric equations of the line tangent to the graph at a given point. Determine if the parametrization is smooth. Find the arc length parametrization given some parametrization.

Vector-Valued functions (VVF): a number a vector component functions of r(t)

COLLABORATE:

The Limit of a VVF:

The Limit of VVF via Component Functions: Continuity of VVF via Component Functions: r(t) is continuous

The Derivative of VVF: Geometric interpretation of the Derivative: The Derivative via Component Functions:

The Integral of VVF via Component Functions:

Basic properties of Differentiation and Integration of Vector-Valued Functions:

Vector-Valued Version of the Fundamental Theorem of Calculus:

Tangent lines to Graphs of Vector-Valued Functions: Derivatives of Dot and Cross Products:

Exercises:

Change of Parameter; Arc Length A Parametrization of C: A smooth parametrization of C: Smooth parametrizations: “Continuously turning” tangent (no cusps)

Example: http://www.math.uri.edu/~bkaskosz/flashmo/parcur/

Example:

Arc Length for Smooth Parametric Curves:

A Change of Parameter: Question: For what g is smooth?

Smooth Change of Parameter: Why? By the Chain Rule:

Arc Length as a Parameter: The “arc length parameter“ s is the signed length of arc measured along the curve from some fixed reference point P and with orientation (s>0 in the “+” direction and s<0 in the “-” direction). The arc length parametrization of C is a parametrization r(s) such that the length of the arc between P and r(s) is |s|. Finding Arc Length Parametrizations: