Tangent conics and tangent quadrics | Differential Geometry 5 | NJ Wildberger скачать в хорошем качестве

Tangent conics and tangent quadrics | Differential Geometry 5 | NJ Wildberger

In this video we further develop and extend Lagrange's algebraic approach to the differential calculus. We show how to associate to a polynomial function y=p(x) at a point x=r not just a tangent line, but also a tangent conic, a tangent cubic and so on. Only elementary high school manipulations are needed--no limits or real numbers-- and we efficiently obtain a hierarchy of approximations to a polynomial at a given point. Instead of derivatives, closely related quantities called sub-derivatives grab the spotlight. They are often simpler and more general quantities! The quadratic approximation, given by the tangent conic, will be crucially important for us in our development of differential geometry: the key point is that the subject is largely what we get when we look at curves and surfaces quadratically! Tangent conics (and higher approximations) of polynomial curves is a potentially rich theory that deserves a lot more attention. We highlight a beautiful observation of E. Ghys: that for a cubic polynomial, the various tangent conics are disjoint (this is in my opinion the loveliest theorem in calculus). Should not all undergraduates be exposed to such natural geometric applications in their calculus courses?? The power of this point of view is shown clearly by the ease with which we can extend it to the multivariable situation. A function of two variables z=p(x,y) defines a surface, which may be studied at a point [x,y]=[r,s] in an analogous way, yielding at each point a tangent plane, a tangent quadric, a tangent cubic surface etc. We explicitly look at the surface associated to the Folium of Descartes, namely z=x^3+y^3+3xy and try to visualize it. This is a powerful but elementary alternative to the usual way of thinking about functions! ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated. Here are the Insights into Mathematics Playlists:    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   Here are the Wild Egg Maths Playlists (some available only to Members!)    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   ************************