MA134 Geometry And Motion Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
When a particle travels across space, it follows a curve. One of the most basic links between geometry and motion may be found here. Because the same curve may be traveled at various, potentially non-uniform, speeds, the motion includes more information than the curve drawn out by the particle (different motion). The integral (with respect to time) of the speed at which the curve is traveled determines the curve’s length (a geometric attribute). The length, on the other hand, seems to be independent of the particle’s actual velocity down the curve. The change of variables formula for integrals is used to establish this independence. The relationship between curvature and acceleration provides another link between geometry and motion.
One learns how to integrate a function of one real variable in high school. This course explains how to integrate vector-valued functions as well as two- and three-variable functions. The area of a surface and the volume of a region (geometry), as well as the circulation of a fluid around a closed curve, will be specified (motion). For two and three dimensional integrals, the change of variables formula will be (heuristically) calculated; it includes a determinant and is somewhat more complex than the one dimensional formula.
Kepler’s Laws of Planetary Motion will be derived from Newton’s second law of motion and the law of gravity in a section on particle mechanics. The basic pendulum’s motion will also be explained. This part builds on MA133 Differential Equations’ examination of gradient flows and introduces the concept of conserved quantities.
The goal of this module is to show students how to convert intuitive geometric and physical notions like length, area, volume, curvature, mass, circulation, and flux into mathematical formulae. It also attempts to educate how to calculate these formulae in practice and how to apply them to basic particle and fluid dynamics issues. In mechanics, the significance of conserved quantities is also emphasized.
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