Differential Geometry : videos 1 35
Differential Geometry is a beautiful classical subject combining geometry and calculus. This is a beginner's course given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. Differential geometry is the application of calculus and analytic geometry to the study of curves and surfaces, and has numerous applications to manufacturing, video game design, robotics, physics, mechanics and close connections with classical geometry, algebraic topology, the calculus of several variables and mostly notably Einstein's General Theory of Relativity.
If your level of mathematics is roughly that of an advanced undergraduate, then please come join us; we are going to look at lots of interesting classical topics, but with a modern, lively new point of view. There will be opportunities for you to contribute to new directions.
Prepare to be surprised, for our approach follows that famous Zen saying:
"In the beginner's mind there are many possibilities; in the expert's mind there are few."
The music is by Exchange: a track called Take Me Higher (thanks Steve Sexton!)
If your level of mathematics is roughly that of an advanced undergraduate, then please come join us; we are going to look at lots of interesting classical topics, but with a modern, lively new point of view. There will be opportunities for you to contribute to new directions.
Prepare to be surprised, for our approach follows that famous Zen saying:
"In the beginner's mind there are many possibilities; in the expert's mind there are few."
The music is by Exchange: a track called Take Me Higher (thanks Steve Sexton!)
This lecture summarizes the basic topics of the course, the unique point of view of the lecturer, and then heads straight into a survey of classical curves, starting with the line, then the conic sections (ellipse, parabola, hyperbola), then moving to classical ways of generating new curves from old ones. These techniques include the Conchoid construction of Nicomedes, the Cissoid construction of Diocles, the Pedal curve construction and the evolute and involute introduced by Huygens. This lecture should be viewed in conjunction with MathHistory16: Differential Geometry.


GeoGebra is a dynamic geometry package, available for free, which allows us to easily make planar geometric constructions which are dynamic (moveable), and investigate associated algebraic formulas and relations. This short lecture gives a brief introduction, since we will be using this software for visualization in this course. We illustrate the program by constructing the ninepoint circle of a triangle.
It is highly recommended that you download the software (free) and play around with it if you do not already have it. 

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 curves, in particular Fermat's cubic, the Folium of Descartes and the Lemniscate of Bernoulli.
We talk about the 17th century's fascination with motion via Newton's laws, and various interesting mechanisms that generating curves, including the four bar linkage and Watt's linkage. 

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 of Analytic Functions (english translation). The idea is to study a polynomial function p(x) by using translation and truncation to create various Taylor approximations to p(x) with respect to a point r on the line. This can all be done with only high school mathematics; in particular NO LIMITS, and NO REAL NUMBERS!! We see that the differential calculus, in its essence, is an elementary theory.
Since this lecture is likely to be of general interest to anyone studying or teaching differential calculus, we begin by reviewing the usual story as found in most modern text books. Then we introduce Lagrange's approach, culminating in identifying clearly the tangent line, tangent conic, tangent cubic etc to a polynomial p(x) at r. These geometric objects associated to a function at a point will play a major role in this course. With this algebraic calculus, the usual derivatives of p(x) are replaced by simpler renormalizations, which we call the subderivatives of p. There are many advantages, and we illustrate some of them with explicit worked out examples. Prepare to have your confidence in the intrinsic rightness of the standard orthodoxy challenged! In this course we adopt a beginner's mind, so there are many possibilities. The implications of the wider mathematical community understanding and appreciating this point of view are huge: maths education can turn a new leaf, with a simpler, more elegant and much more logical approach to this important subject. As Euler and Lagrange tried to teach us, more than 200 years ago. nsw.edu.au/~norman/. 

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 neededno 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 subderivatives 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! 

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 trying to use a two dimensional representation to understand a surface in three dimensions, with its tangent planes and tangent quadrics.


With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. One big difference here is that the subderivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving subderivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. This lecture shows that the calculus is actually a much wider operational tool than is usually appreciatedfinite calculus not only makes sense but is a rich source of both combinatorial and algebraic patternsand questions for further investigations. 

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 and so on to a general curve of the form p(x,y)=0; this includes the situation y=f(x) as a special case. It also allows us to deal with situations where the usual tangent is vertical, and so the derivative is undefined.
The case of the lemniscate of Bernoulli is looked at in detail. Since now the tangent conic can be either an ellipse, parabola or hyperbola, we see that the nature of the quadratic approximation at a point allows us to group points on the curve into elliptic, parabolic and hyperbolic type. For the lemniscate, the parabolic points are found, lying on the discriminant conic. This opens the door to a more differential study of algebraic curves. We also show how this strategy generalizes in a simple and natural way to investigate algebraic surfaces in three dimensional space! 

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 considered as the space of onedimensional subspaces of a three dimensional vector space, or in other words lines through the origin. In this way we can introduce homogeneous coordinates [X:Y:Z] for the more familiar points [x,y]; the big advantage is that now points at infinity become concrete and accessible: they are simply points of the form [X:Y:0].
A curve like the parabola y=x^2 gets a homogeneous equation YZ=X^2, including now the point at infinity [0:1:0], which corresponds to the direction in the y axis. This gives a uniform view of conics close to Apollonius' view in terms of slices of a cone. We will see that homogeneous coordinates provide a powerful and useful tool to not only the study of conics and algebraic curves in the plane, but also to quadrics and higher algebraic surfaces in space. 

Projective geometry is a fundamental subject in mathematics, which remarkably is little studied by undergraduates these days. But this situation is about to changethere are just too many places where a projective point of view illuminates mathematics. We will see that differential geometry is no exception.
In this video we show how to view a general conic in a projective way, yielding an important correspondence with 3x3 symmetric projective matrices. This motivates the introduction of projective linear algebra: where the basic objects are invariant under scaling. We distinguish between projective points as row vectors, projective lines as column vectors, and incidence in terms of the usual matrix product between them. This brings out the all important duality in projective geometry, usually missing from most linear algebra treatments, where the affine view obscures the symmetry between points and lines. We explain Pappus' theorem from this view, and show its dual result. We then have a look at some advantages in representing points and lines in the plane with projective coordinatesfirst of all we can use the third coordinate to absorb denominators, meaning that fraction arithmetic can be replaced by integer arithmetic. Secondly the main computations of finding meets and joins both reduce to a single computation: finding a cross product of two vectors. 

We review the simple algebraic setup for projective points and projective lines, expressed as row and column 3vectors. Transformations via projective geometry are introduced, along with an introduction to quadratic forms, associated symmetrix bilinear forms, and associated projective 3x3 matrices. An important example is the Lorentz/Einstein/Minkowski geometry. Then notions of perpendicularity are closely related to the polepolar duality between points and lines associated to the unit circle in the plane. This goes back to Apollonius, and is closely related to the developments in the UnivHypGeom series.


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 crucially important concept of curvature, first of all for a parabola of the form y=ax^2.
Metrical structures are usually associated with quadratic forms, or the corresponding symmetric bilinear forms, also known as dot products or inner products, and are represented mathematically by symmetric matrices. These are also naturally linked to conics. The essential equivalence between these seemingly different objects is an important understanding. For applications to physics, in particular relativistic geometries, it is useful to have a flexible attitude here about metrical structurethe Euclidean one is not the only one to consider. This is rather a departure from classical courses in differential geometry, but the student will gain much by adopting such a more flexible and general point of view. We discuss twodimensional relativistic geometries, and explain why circles in these contexts are what we would usually call rectangular hyperbolas. For those particularly interested in these metrical connections between 2D and 3D relativistic geometries, I suggest the video in my MathSeminars series called Hyperbolic Geometry is Projective Relativistic Geometry. In the last part of the lecture we begin studying the parabola y=ax^2 at the point [0,0], with a view of determining the relations between the focus of the parabola, the center and radius of curvature, and the curvature itself. We are following here the approach of Zvi Har'El, explained in a paper called: Curvature of Curves and Surfaces: a Parabolic Approach (1995). 

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 defined projectively as a conic which is tangent to the line at infinity.
We find the general projective 3x3 matrix for such a parabola with its vertex at [0,0]. We then derive the focus directrix definition of such a parabola and study its local behaviour at the origin, in particular connecting it to a function representation of the form y=alpha x+beta x^2 +... This lecture has some simple but no doubt unfamiliar formulas which pin down the curvature of a general parabola: the main ingredient to understanding curvature of a general curve, following the philosophy of Zvi Har'El in basing curvature on the approximation by parabolas. 

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 basing things on the parabola is that we get nice connections between curvature and the foci, and that once we move to studying surfaces in three dimensional space, the normal paraboloids will play an exactly analogous role as do the normal parabolas here.
This lecture does have quite a lot of formulas in it, because we are interested in laying out technology that will easily allow us to move from one view to another. At some point, we begin to appreciate the power of formulas in pinning down this subject, but it does help to develop some sympathy with algebraic relations! A point of divergence from the usual formulation: we emphasize the usefulness in thinking about the square of the usual curvature. This quadratic curvature can be formulated without any appeal to square roots. Correction: That equation at 18:58 should have the factor (mxly)^2, not (lxmy)^2 (it can be properly remembered by being a cross product term). Following on, the next equation should also have the same correction. Thanks Thomas Fuhrmann! 

We continue in developing fundamental formulas that deal with curvature for surfaces in three dimensional space, given by algebraic equations. Our approach continues to be that the normal paraboloid to such a surface at a point is the key object that encodes the quadratic metrical information, including curvatures. However we want formulas that deal with the general situation, not only the simpler case when the tangent plane is horizontal.


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 about irrational numbers and transcendental functions: which we treat from an applied point of view: all statements involving integrals, infinite sums etc are to be interpreted in an approximate sense. (For those interests in finding out more about such radical departures from the established dogma, see my MathFoundations series!)
We treat curvatures from a rational turn angle point of view, introduced in the AlgTop series. 

We look at extremely big numbers. This is the best way to get a feel for the immensity and complexity in the sequence of natural numbers. And why we have no right to talk about `all' of them as a comaWe continue the discussion of planar curvature of a curve, including the notion of turning number and winding number. This connects with the notion of the degree of a map from a circle to a circle, and relates also to a useful result in complex analysis sometimes called the Dog on a Leash theorem.
Then we extend the discussion to curves given parametrically, starting with the important example of uniform motion on a circle, where the classical circular functions and their inverses play an important role, and discuss reparametrization to the standard case of a unit speed curve.pleted `infinite set'. Our main tool is a cool inductive way of defining higher and higher operations, going beyond multiplication and exponentiation. 

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 the language of vector valued derivatives.
The idea is to attach to each point of the curve, a triple of unit vectors, called traditionally T, N and B, the tangent, normal and binormal unit vectors. These form at every point a mutually perpendicular frame of basis vectors, much like the i,j and k standard unit basis vectors along the x,y and z axes. The vector T is in the direction of the curve (T standards for tangent), while N is in the direction of the acceleration, which for a unit speed curve must be perpendicular to the tangent. The third vector B can be defined as the cross product of T and N. The Frenet Serret equations describe what happens as we move along the curve with unit speed s, namely what are the derivatives of T,N and B with respect to s. The curvature k(s) comes into play, as does a new quantity called the torsion, usually denoted tau(s). 

Following from the last lecture on the Frenet Serret equations, we here look in detail at an important illustrative examplethat of a helix. The Fundamental theorem of curves is statedthat the curvature and torsion essentially determine a 3D curve up to congruence.
We introduce the osculating, normal and rectifying planes, and try to explain the physical meaning of torsion. 

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 plane, while the span of adjacent normals (or equivalently the velocity and acceleration) is the osculating plane.
In this lecture we start by continuing on from the example of the last lecturediscussing curvature and torsion for a finite segment curve, which is useful in robotics. Then we go back to G. Monge who introduced geometrical aspects of space curves, such as the idea of the axis, also polar developable and tangent developable surfaces associated to a space curve. An example for the rational helix is illustrated. We are also interested in algebraic aspects of curves, and give alternate formulas for the curvature and torsion of a curve in terms of the velocity and acceleration. We are particularly interested in the general case where we do not assume a unit speed curve. We also study how invariants may be considered as quantities which are invariant under reparametrization of the curve. 

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 then discuss the division in the subject between algebraic surfaces and parametrized surfaces. It is very important to have a balanced view between these two aspects; most texts are oriented, following Gauss, to the parametrical side: we will at least initially compensate by providing more detail on the algebraic surfaces.
Important examples of surfaces include quadrics, such as spheres, or more generally ellipsoids, or hyperboloids. We also mention some more unusual examples, including the Oloid, discovered by Paul Schatz in 1929. 

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 particular conic x^24xy2y^2=3, and finding the evolute of the curve y=x^n for a general n.
Note that in my diagram around 14:00 I incorrectly place the point [1,1], but this has no effect on the algebraic discussion. 

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 of a collection of open sets. While this is all reasonably traditional stuff, regular viewers of this channel will not be surprised to learn that I consider the `infinite set' aspect of these theories highly dubious.
But this is a course at a major university in 2013, so it is still early days for people to be looking more critically at such theories. Of course there are finite versions of these concepts, and we do use some simple examples for illustration, but it is fair to say that these do not capture the true intention of these definitions in trying to set up a theory of what a `continuous space' might actually be. 

Paraboloids are going to play a special role in our understanding of curvature. The idea is that we are going to locally approximate a surface S near a point by a normal paraboloidone that shares the same tangent plane, but has an axis which is perpendicular to that tangent plane. It will turn out that the curvatures of this normal parabola can then be used to define the curvatures of the surface. This is a very flexible and useful approach to curvature.
In this lecture we start a discussion of paraboloids, which are three dimensional analogs of parabolas. They are closely related to the algebraic notion of a quadratic form, or equivalently a symmetric bilinear form. Studying such things involves some linear algebra which we review. In particular we discuss diagonalization of quadratic forms, eigenvalues and characteristic polynomials. Dupin's indicatrix makes an appearance. 

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 of a collection of open sets. While this is all reasonably traditional stuff, regular viewers of this channel will not be surprised to learn that I consider the `infinite set' aspect of these theories highly dubious.
But this is a course at a major university in 2013, so it is still early days for people to be looking more critically at such theories. Of course there are finite versions of these concepts, and we do use some simple examples for illustration, but it is fair to say that these do not capture the true intention of these definitions in trying to set up a theory of what a `continuous space' might actually be. 

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 associated with specification, with the use of `infinite sets', with the notions of `functions' etc.
Even those students who aspire to understanding mathematics correctly ought to be at least aware of the standard formulations, and if one is teaching a course at a major university these days one is limited by the curriculum and the orientation of students and other lecturers in the level of directness that one may address these foundational problems. I will eventually be discussing the difficulties with these concepts in the MathFoundations series. In this lecture we talk about charts, manifolds, orientation, and then look more carefully at the two dimensional case of compact surfaces, where things are more concrete and explicit, largely through the classification of Dehn and Heegard which utilizes in a major way the Euler characteristic. 

We describe the important classification of compact, oriented 2manifolds, and the relation with the topological invariant called the Euler characteristic. The idea is to work combinatorially, by decomposing a 2manifold into polygon pieces which are glued, or identified, along common edges, and then performing cut and paste operations to try to get such a configuration into a normal form.
This was worked out by Dehn and Heegaard around 1910, and the resulting classification is one of the most important foundational results in differentential geometry and algebraic topology. In fact they worked in a more general context which also includes nonorientable surfaces, such as the projective plane or Klein bottle, but for differential geometry it suffices to work with orientable surfaces, which have a consistent notion of positive orientation on small circles around the surface. Remarkably, the resulting classification shows that the Euler invariant completely determines the compact oriented surface. NOTE: Due to a mistake in numbering, there is no DiffGeom27 video! So please skip ahead to DiffGeom28. 

Here we introduce a somewhat novel approach to the curvature of a surface. This follows the discussion in DiffGeom23, where we looked at a paraboloid as a function of the form 2z=ax^2+2bxy+cy^2.
In this lecture we generalize the discussion to the important case of a paraboloid, which we define projectively as a quadric which is tangent to the plane at infinity, and whose vertex has been translated to go through the origin. We show how to represent such a paraboloid algebraically using a projective 4x4 matrix., and write down the corresponding Cartesian equations. Such a quadric will have two curvatures at the origin: both are defined in terms of the coefficients of the characteristic polynomial, suitably renormalized so they are independent of scaling. The first curvature K_1 is up to a factor of 4 the square of the usual mean curvature, while the second curvature K_2 is the usual Gaussian curvature. So we see that curvature is really coming from linear algebra and the characteristic polynomial of a symmetric matrix. This is an important insight that lends itself pleasantly to higher dimensional generalizations. 

We extend our approach to curvature to general algebraic surfaces. The formulas get involved, but they have pleasant symmetry and are quite powerful.


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 special cases. The formulas were given in several roughly equivalent forms, applying to different situations. The first applied to a normal paraboloid: a paraboloid which has a vertex at the origin with a specific tangent plane there. The second applies to a function z=f(x,y) which passes through the origin. Then the third was for the general conic passing through the origin.
In each case linear algebra and the characteristic coefficients yield two curvatures, K_1 and K_2. The first curvature K_1 is close to being the square of the mean curvature classically, but there is not a factor of two, and the second curvature K_2 is the Gaussian curvature. Our examples begin with the sphere, and then move to an ellipsoid. Then we discuss the ellipson, a cubic surface coming from Rational Trigonometry (this is perhaps the most important cubic surface!) After that we examine the Ding Dong surface. Clearly there is a lot of potential for investigating other surfaces, including cubic surfaces. Amateur mathematicians and studentsthere is a lot of scope here for interesting calculations and generalizations!! Roll up the sleaves, and get to work, if you like! We end the lecture with a discussion of curvatures for parametric surfaces, in the direction of C. F. Gauss. 

We use vectors to introduce parallelograms, the parametric representation of a line, and affine combinations, such aaThis 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 was a soldier, engianeer and mathematician. He investigated lines of curvature and discovered a famous result that shows how to compute the sectional curvature of a surface cut by a nonnormal plane, which we restate using the language of Rational Trigonometryof course!
We discuss also elliptic, hyperbolic and parabolic points and umbilic points. 

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 subject. He was the inventor of descriptive geometry (which he developed for military applications), and various theorems in Euclidean geometry, including homothetic centers of three circles, and the Monge point of a tetrahedron. He also studied curves, families of surfaces, edges of regression, and lines of curvature.


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 surfaces and lines of curvatures, for example confocal families of ellipses and hyperbolas. He studied conjugate directions on surfaces (going back in some sense to Apollonius), and introduced the Dupin indicatrix, which is closely related to our strategy of using a normal approximating paraboloid to define and compute curvatures.


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 oriented surface S to the unit sphere S^2, which describes how a unit normal moves along the surface.
The first fundamental form describes the Euclidean quadratic form in terms of the parametrization, and the second fundamental form is determined by the derivative of the Gauss  Rodrigues map. 

In this video we discuss Gauss's view of curvature in terms of the derivative of the GaussRodrigues map (the image of a unit normal N) into the unit sphere, and expressed in terms of the coefficients of the first and second fundamental forms. We have a look at these equations for the special case of a paraboloid, where we can compare with our previous discussion of curvature.
We then look at a discrete analog of curvature which applies to polyhedra, which goes back to Descartes. Involved are formulas for the sum of angles of a spherical polygon. This discrete form gives us an easy justification of Gauss' Theorema Egregium. We have a look at the Gauss Bonnet theorem in this context, where the total curvature of a closed surface with the topology of the sphere is 2 pi times the Euler characteristic. This is the final video in this series (but we may very well carry on at some future point!) 
