MA 242.003 Day 13- January 24, 2013 Chapter 10, sections 10.1 and 10.2.

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Presentation transcript:

MA Day 13- January 24, 2013 Chapter 10, sections 10.1 and 10.2

Space curves DEFINITION: A space curve is the set of points given by the ENDPOINTS of the Vector-valued function when the vector is in position vector representation.

An example:

Another Example:

My standard picture of a curve:

Section 10.2: Derivatives and Integrals of Vector-valued functions

Section 10.2: Differentiation Rules See the boxed theorem in section 10.2 on Differentiation Rules

Section 10.2: Differentiation Rules See the boxed theorem in section 10.2 on Differentiation Rules In particular we will need:

Section 10.2: Differentiation Rules See the boxed theorem in section 10.2 on Differentiation Rules In particular we will need:

Maple’s vector calculus spacecurve Tutor under Tools

Integration of vector-valued functions

Example:

Section 10.3 Arc Length and Curvature To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve.

Section 10.3 Arc Length and Curvature To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve. Compare the unit tangents for – 1. a straight line

Section 10.3 Arc Length and Curvature To describe the acceleration r’’(t) it turns out that the crucial idea is CURVATURE of the curve. Compare the unit tangents for – 1. a straight line – 2. a curved line

Curvature of a straight line is then ZERO

Curvature of a non-straight line is then NON-ZERO

Curvature of a straight line is then ZERO Curvature of a non-straight line is then NON-ZERO Problem: The number for the curvature depends on choice of parameter.

My standard picture of a curve: