Multivariable Calculus: Velocity and Acceleration

A closer look at r'(t) and r''(t) for parametrized curves. Using the unit circle as an example, we explore how different parametrizations of the same curve can yield varying properties like speed and acceleration. The lecture highlights the significance of velocity and acceleration vectors, their relationship to the curve's geometry, and how they change under reparametrization. Through mathematical computations and a MATLAB demonstration, we illustrate these concepts, culminating in an understanding of the orthogonality between velocity and acceleration in constant speed parametrizations. (Unit 2 Lecture 4)
Key Points
1. Unit Circle Parametrization: Analyzed the unit circle's standard parametrization and its properties.
2. Velocity and Speed: Explored the velocity vector's properties and constant speed in the standard parametrization.
3. Acceleration Vector: Examined how the acceleration vector relates to the curve and its position vector.
4. Reparametrization Concept: Introduced the idea of reparametrizing a curve and its implications.
5. Velocity, Speed, and Acceleration in Reparametrization: Computed and compared these quantities in a new parametrization of the unit circle.
6. Visualization: Demonstrated the differences in parametrizations using MATLAB.
7. Orthogonality of Velocity and Acceleration: Proved that in constant speed parametrizations, velocity and acceleration vectors are orthogonal.
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