An algebraic infinitesimal approach to product and chain rules | FMP 22c | N J Wildberger

In this video we give the algebraic framework for a general infinitesimal approach to the Derivative of Faulhaber, valid for a general field F. We rely both on the notion of a dual complex number over F, and the idea of a bi-polynumber. Dual complex numbers are 2 x 2 matrices that incorporate and algebraic infinitesimal epsilon. Bi-polynumbers are data structures that capture the idea of a polynomial function of two variables, but crucially the notion of "variable" here is replaced rather by specific data structures.

With these preliminaries, we introduce the important Derivative theorem. This is a somewhat novel way of stating exactly what it is that happens when we take a Derivative, but without limits, and without "real numbers". Then we derive the Product Rule and the Chain Rule, again in a very general context.

These important directions will be developed at greater length and with lots of examples in the Algebraic Calculus Two online course.

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[bernardoxbm]

Happy new year! In my opinion your are the best Math teacher. Thank you for sharing!

[federicofresneda]

Still the best mathematician on YouTube. Happy new year and thanks for your work.

[tricky]

This is superbly interesting stuff! Thank you!

[mahiainti]

It's interesting how "naive" derivative definition D[p]=(p(x+dx) - p(x))/dx turns out to have such beautiful math behind it!

[peterosudar]

It is so interesting to think about how calculus over a finite field could lead to better appreciation for how chromogeometric principles will begin to become more obvious in their importance in defining notions that "curriculum" all over the world attest to "teaching" in elementary, secondary and post-secondary education institutions.

Pretty obvious, to me, that engaging with "cubes" will become essential to "developing rigour" in one's "mind's eye" about grasping with mathematical content over the course of on'e journey with life-long learning of mathematics...

While I don't think we will see this occur in 2021... we might be ready for mainstream with this ideas in.... 2030? Maybe?

Until then... Thank you for inspiring us to think deeply about mathematics.

[jarogniewborkowski]

You gave me a lot of inspiration to reformulate physics laws into new, correct Mathematical language. But I need Your help with Differential Equations. Could You make a video about Algebraic Differential Equations and show how to write and solve them in new Math-language? Best regards

[Rudi_F_Vienna]

Dear Norman, your approach reminds me very much of the somewhat more general exposition in van der Waerden's "Algebra". He considers polynomials of the form p(x+h) in the polynomial ring R[x, h], where R is an arbitrary commutative ring, x corresponds to your alpha, and h corresponds to your beta*epsilon. Then p(x+h)=p(x)+p'(x)*h (mod h^2), which corresponds to your derivative theorem. Sum rule, product rule and chain rule can be proved in the same way as you do, but always working mod h^2. Working mod h^2 has the same motivation as your choice of epsilon as a matrix that squares to the null matrix. It seems to me that up to now your results cover the special case with R=C_d, where C_d is the ring of the matrices that you call the dual complex numbers over a commutative field, which of course includes all finite fields. I am looking forward to your next lecture!

[reptilewithsadhumaneyes]

I’d like to see a substitution rule proof. Since alpha and beta are an orthonormal basis in the usual sense, it feels like there may be a counterexample of some identity involving alpha and beta that doesn’t hold for another two poly numbers. Maybe I am missing the definition of what is considered an identity.

[pickeyberry]

Video Content

00:00 Introduction
01:59 Faulhaber Derivative
04:12 Bi-polynumbers over IF
07:36 Multipilication is a 2-D cauchy product
12:29 Bi-polynumber(IF) is a commutative algebra
16:55 Derivative theorem
20:32 Product Rule
22:44 Substitution Rule
25:10 Chain Rule

[derendohoda]

I love thinking about extension fields and rings and such. Knew about dual numbers and their connection to derivatives for a long time. I think doing algebra on dual numbers (not a field, alas) and field extensions of rationals gives some mechanical insight as to why the real numbers are so insane. I would add to this doing hand calculations with p-adic rationals and negative p-adic numbers. Trying to leave the rationals even a little bit, everything gets extremely weird.

[Igdrazil]

Precision : In a POLYNOMIAL the illicitely so called "variable" IS NOT actually a "variable", but simply an ABSTRACT BASIS, eventually NUMERICAL just as 10 in 234=2.10^2+3.10+4. We abusively talk about the "variable" x, since we often shunt to the polynomial associated FUNCTION when the context is numerical.

An other proper way of looking at a POLYNOMIAL is as a VECTOR in the (1, x, x^2, ..., x^n) vectorial ABSTRACT BASIS, that can be linked one to one (through a homomorphism) to n-dimentional CANONICAL vector BASIS, made of (1, 0, 0...), (0, 1, 0, ...), ...(0, 0..., 1), which is A HARD CORE "ATOMIC DATA".

So the concept of "variable" actually arises when talking not about polynomial, but POLYNOMIAL FUNCTIONS, which is a map that asignes the value, for instance 2x^2-3x+6 to each numerical value given to the "variable" x. But the variable need not to be a classical number, it can be any generalised one : a complex number, a dual complex number, a matrix, etc, as long as the main arithmetic operations can be performed onto in a closed arithmetic well defined arena.
And the FUNCTION concept is very clearly defined as a TRIPLET (A, B, G) of INPUTS of A type, OUTPUTS of B types, and a GRAPH G wich is a part of the CROSS PRODUCT A*B. Eventually a FUNCTIONAL type graph, that means "one to one". But MULTIVALUED "FUNCTIONS" with non "functional graphs" are aslo very interesting and important mathematical objetcs, alike Riemann surfaces ("square root of z in complex number kingdom) that alows to transform a multivalued graph in a singled valued one, as soon as the initial "R^n" representation is replaced by a wiser one, on a so called a manifold.

So POLYNOMIAL concept is much wider and more abstract, thus general, than polynomial numerical functions, that are just one aspect of polynomials. Even polynumber is only one aspect of POLYNOMIAL, where the "coefficients" happen to be FIELD NUMBERS. Polynomial alows to deal for instance with chains of characters or symbols, as long as one defines the precise meaning in such context to addition, scalar multiplication and multiplication. It can be made of CONCATENATION for instance and other creative stuffs.
So classical numbers are not the limit of POLYNMIAL kingdom, which is much wider.

In these regards, POLYNOMIAL is really a perfect KEY CORNER STONE concept, extremely universal, and openly multi-"variabled". If ones look at it properly, it can also grab chinese calligrams that are square matrix blocs, filled with basic keys. With a few hendred keys, all 50000 chinese words can be built. It's just Bi-polykey 2*2 matrices or two "variable" polynomial.

[sriharshanuthalapati]

Hi, I watched few of your videos, it made me thinking how our life was made simple by having complex numbers, existence of infinity sequence. I have a question on your opinions about analytic continuation that happen in real world physical situations like quantum tunneling and evanescent wave that appears when there is a Total Internal Reflection of Light. When solving these, we often use complex numbers in our equations.

[Igdrazil]

Key remark : in this chain rule formula, Norman have found a way to separate by the dual complex number algebra, the "MANIFOLD" PART wich is expanded on the unitary matrix vector basis, and the "TANGENT SPACE" PART wich is expanded on the EPSILON matrix vector basis. All writen in a unifying algebra, without no more amalgamic usual confusion of ordinary calculus. Indeed when ordinary calculus writes df=f'(x)dx, and f(x+dx) = f'(x)+f'(x)dx+ etc, the dx is actually NOT AT ALL an "infinitesimal", as physicists love to look at it (as an arbitrarely "small quantity"), but actually a LINEAR ONE-FORM EATING VECTORS OF THE TANGENT SPACE.

So the mix writing of f(x), which is a number, side by side f'(x)dx, which is actually a one-form, is heavely confusing for 99% of Physicists and lots of mathematicians to. In all this road, everything is writen in a particular CHART (of two "variable" coordonates x and y), but the true meaning of dx or dy is a one-form basic vector of the COTANGENT BUNDLE, eating vectors of the TANGENT BUNDLE. It does by NO MEAN represent any "infinitesimal" quantity. Nevertheless Physicists use it as such and Cracken so many concrete hard problems with this point of view.

So this dual complex number algebra offers a sort of very accessible and easely understandable RECONCILIATION of these two apparent opposite points of views. The epsilon matrix is a chirurgical knife to extract the TANGENT/COTANGENT information of the concerned MANIFOLD.

For instance lets look at S1 (manifold) circle, immersed and embedded in "R2", defined by its algebraic aquation x^2+y^2=1. In traditional Calculus, one has the ugly task of first puting it in functional form, given thus by two functional irrational equations : y=+sqrt(1-x^2) (uper half circle) and y = -sqrt(1-x^2) (lower half circle). Then to compute the derivative of each : y'=-x/sqrt(1-x^2) and y'=-IDEM. This allows to compute the differential one form dy expanded on the basic one-form dx : dy=y'.dx. With and additional work to reconstruct the geometric "tangent vector" in this "R2" embedding, at any point (x, y) of this S1 manifold, namely the TANGENT VECTOR (1, y'), or alike (dx, dy), both calculated at (x, y) point of the circle.

This heavy and often impossible task of reframing the initial problem in functional form, can be overcome by INTRINSIC CALCULUS of differential forms. The direct and straight forward DIFFERENTIATION of the algebraic circle equation gives immediately : 2xdx+2ydy=0. On an intrinsic point of view, this shows that the two one forms dx and dy are not linearly independant. Moreover, on an extrinsic point of view of charts of "R2 flat space" with the usual euclidian scalar product, this differntial equation shows that the RADIAL VECTOR (x, y) is always ORTHOGONAL to the TANGENT one (dx, dy). So here again we mix the one-form formalism and the infinitesimal story, in a Janus way. Strictly speaking, from a (differential) one-forms point of view, dx does not represent an infinitesimal, but a cotangent one form eating one dimentional "tangent vectors". The other one-form dy being an alternative to dx, but here not independant of dx, as long as we stick on the manifold S1. The reason of this dependancy lies in the fact that the euclidian embedding by the two variable chart to map S1, is surnumerous, since S1 is essentialy a one dimensional manifold. Topologycally, intrinsically, the S1 circle has no particular shape, nor "round" form, it is just a one-dimentional manifold of gender 1, with one hall, thus connected, but not simply connected. Any loop making more than one turn, canot be shrinkened to zero by continuous means. A more classical and elementary way of seeing that is to notice that usual polar angle theta can perfectly map S1, in his usual "round" shape x^2+y^2=1 or in any topological weird shape. The so called "curvilinear abscisse" is an alternative one dimentional chart. Norman has been given some thaughts on the intrinsic definition of the unit circle, watch the corresponding video on that topic, very interesting.

So now, third tool to discribe S1 : the dual complex purely algebraic formalism with EPSILON "infinitesimal" nilpotent matrix. Who writes this down? ;)

[MichaelKolczynski]

The substitution rule is confusing. Only because I don't currently understand its necessity. Alpha is a polynumber. Beta is a different polynumber. That alpha does not equal beta is fine. But as written both example expressions are duplicates but for character swapping - so there isn't any additional information.

It feels too obvious to need explaining that the explanation is confusing due to its lack of justification for its inclusion.

It's not a proof but also not a definition.

Almost an aside but given far too much time for an aside.

Maybe it's due to lack of ultimate explicity over how two polynumbers in alpha may not be equal but a polynumber in alpha and a polynumber in beta with identical structure are not equal in a different way.

a^3 is not equal to a^2 but a^3 is also not equal to b^3.