We introduce surfaces, which are the main objects of interest in differential geometry. After a brief introduction, we mention the key notion of orientability, and ...
This lecture relates the two dimensional surfaces we have just classified with the three classical geometries- Euclidean, spherical and hyperbolic. Our approach ...
Projective geometry is a fundamental subject in mathematics, which remarkably is little studied by undergraduates these days. But this situation is about to ...
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) ...
This lecture is an introduction to knot theory. We discuss the origins of the subject, show a few simple knots, talk about the Reidemeister moves, and then some ...
With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, ...
In this video we discuss Gauss's view of curvature in terms of the derivative of the Gauss-Rodrigues map (the image of a unit normal N) into the unit sphere, and ...
The first lecture of a beginner's course on Differential Geometry! Given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW.
Here we give an informal introduction to the modern idea of `manifold', putting aside all the many logical difficulties that are bound up in this definition: difficulties ...
This is the first of three videos that discuss the mathematical lives and works of three influential French differential geometers. We begin with J. Meusnier, who ...
This video presents a summary of classical spherical trigonometry. First we define spherical distance between two points on a sphere, then the angle between ...
We introduce the notion of topological space in two slightly different forms. One is through the idea of a neighborhood system, while the other is through the idea ...
Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the ...
This is the fourth lecture of this beginner's course in Algebraic Topology given by N J Wildberger of UNSW. This lecture continues our discussion of the sphere, ...
In this tutorial we explore the surface z=x^3+y^3+3xy using GeoGebra. The aim is to develop our skills using this dynamic geometry package, at the same time ...
The central theorem in algebraic topology is the classification of connected compact combinatorial surfaces. In this lecture we introduce this result and indicate ...
In this lecture we introduce a general approach to metrical structure, via a symmetric bilinear form in either an affine or projective setting, and then begin studying ...
The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with ...
We rejuvenate the powerful algebraic approach to calculus that goes back to the work of Newton, Euler and particularly Lagrange, in his 1797 book: The Theory ...
We look at some of the work of Charles Dupin, a French naval engineer and student of Monge. He made some lovely discoveries about triply orthogonal ...
A first look at Projective Geometry, starting with Pappus' theorem, Desargues theorem and a fundamental relation between quadrangles and quadrilaterals.
This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on ...
In this video we introduce projective geometry into the study of conics and quadrics. Our point of view follows Mobius and Plucker: the projective plane is ...
GeoGebra is a dynamic geometry package, available for free, which allows us to easily make planar geometric constructions which are dynamic (move-able), ...
Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Projective geometry is ...
In this video, I introduce Differential Geometry by talking about curves. Curves and surfaces are the two foundational structures for differential geometry, which is ...
We now extend the discussion of curvature to a general parabola, not necessarily one of the form y=x^2. This involves first of all understanding that a parabola is ...
Red geometry is a two dimensional relativistic geometry in the spirit of rational trigonometry, using variants of the usual quadrance and spread. The usual grid ...
The Frenet Serret equations describe what is happening to a unit speed space curve, twisting and rotating around in three dimensional space. This is done with ...
In the 1930's H. Siefert showed that any knot can be viewed as the boundary of an orientable surface with boundary, and gave a relatively simple procedure for ...
We introduce the approach of C. F. Gauss to differential geometry, which relies on a parametric description of a surface, and the Gauss - Rodrigues map from an ...
In this video we extend the discussion of curvature from parabolas to more general conics, and hence to more general algebraic curves. The advantage of ...
Following from the last lecture on the Frenet Serret equations, we here look in detail at an important illustrative example--that of a helix. The Fundamental ...
We discuss the curvature of planar curves and applications to turning numbers and winding numbers, also called the index. We use this opportunity to talk a little ...
Here we continue our study of the works of three important French differential geometers. Today we discuss G. Monge, who is sometimes called the father of the ...
A space curve has associated to it various interesting lines and planes at each point on it. The tangent vector determines a line, normal to that is the normal ...
We look at terminology and notation associated to curves, in particular the classical conic sections, functions such as the exponential and logarithm, and some ...
This lecture relates the two dimensional surfaces we have just classified with the three classical geometries- Euclidean, spherical and hyperbolic. Our approach ...
This video follows on from DiffGeom21: An Introduction to surfaces, starting with ruled surfaces. These were studied by Euler, and Monge gave examples of how ...
This is the 5th lecture of this beginners course in Algebraic Topology. We introduce some other surfaces: the cylinder, the torus or doughnut, and the n-holed ...
In this video we extend Lagrange's approach to the differential calculus to the case of algebraic curves. This means we can study tangent lines, tangent conics ...
After the plane, the two-dimensional sphere is the most important surface, and in this lecture we give a number of ways in which it appears. As a Euclidean ...
This lecture discusses parametrization of curves. We start with the case of conics, going back to the ancient Greeks, and then move to more general algebraic ...
How to think about both projective points and projective lines via lines and planes in 3D geometry. Also we discuss some basic facts about 3D geometry, relating ...
We review the formulas for the curvature of a surface we derived/discussed in the last lecture, and then give explicit examples of how these formulas work out in ...
This video gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler ...
We describe the important classification of compact, oriented 2-manifolds, and the relation with the topological invariant called the Euler characteristic. The idea ...
Here we go over in some detail three problems that were assigned earlier in the course: the rational parametrization of the cissoid, the parametrization of a ...
We review the simple algebraic set-up for projective points and projective lines, expressed as row and column 3-vectors. Transformations via projective ...