I highly disagree with the resolution of the "notation problem" at 19'13". The problem is well stressed, but ill solved, if a cat ought to be called a cat! A little "mess" indeed is runing around there, in the lecture notation, and confusion is rising with paradox from a rather different cause than the one stressed out, if RIGOUR is to be called RIGOUR!The little "mess" that is at the core of these confusions is simply that, through the change of variables, though f(x,y) = x+y = 2u+v = g(u,v), which are NUMERICAL EQUALITIES and which are NUMERICALY true, nevertheless the functions f and g ARE NOT the same sincef : (x,y) --> f(x,y)=x+y, and g : (u,v) --> g(u,v) = 2u+v, WHICH IS equivalently written (since variables are "DUMMY VARIABLES") g : (x,y) --> g(x,y) = 2x+y.From this rigourous functional (or algorithmical) way of looking at it,  it's obvious that f and g are DIFFERENT algorithms, that is DIFFERENT functions!With things clarified this way, all confusions, danger, paradox... simply vanish : (df/dx)y = 1, and (dg/du)v=2 No more pb!...More over, be now clearly aware that we can write, with absolute rigour : (dg/du)v = (dg/dx)y, BUT ONLY IF YOU READ THIS IDENTITY IN THE PROPER WAY ONE SHOULD READ IT, THAT IS THROUGH "FUNCTIONS SPIRIT", AND NOT "number spirit". Why that? Because variables in these expressions, ARE AND MUST BE LOOKED AT, AS "DUMMY VARIABLES"! More precisely, what both of these expressions (dg/du)v or (dg/dx)y, trully mean, is simply and unambiguously : DIRIVE WITH RESPECT TO THE FIRST VARIABLE (OF THE ORDERED PAIR, OF THE CROSS PRODUCT OF SETS), KEEPING THE SECOND "VARIABLE" CONSTANT. Therefore we could equivalently have written (dg/dk)@  (or any crazy other "names" of variables), if we had defined the function "in such CHART", labeled (k,@),  by :g : (k , @) --> 2k+@THIS is the only way of puting things trully clearly and rigourously, without ever giving rise to any notation confusion or paradox! Simply by calling a function a function, and NOT "a cat a dog", messing around with words and concepts, through rather "physicist" lasy notations wich turns sometime to prove to be "bad" habits. It's in fact a CHAIN RULE MISTAKE that is arising when one misleadingly identify f with g. If one realy wants to differentiate with respect to the same variable, x for instance, keeping y constant, it is possible! But to perform that without any mess and mistake, one SHOULD NOT naively compare  (df/dx)y=1 with (dg/du)/v=2, wich are clearly DIFFERENT (and nothing wrong about it!). But instead, one SHOULD think "functionaly" instead of "numericaly"! That means one has to compute clearly the "CHAINED FUNCTION" G, which is different from g, but which is related to g by the Change of Variable. We thus express, first the change of variable in the PROPER WAY! That is, as the following FUNCTION :C : (x,y) --> C(x,y) = (u,v) = (x, y - x)We thus recall that the FUNCTION g is defined as the following algorithm :g : (@,#) --> 2@ +#,or equivalently (since what any function trully IS, IS it's defining ALGORITHM!), but more conveniently (for our CHAINING FUNCTIONS GOAL)  :g : (u,v) --> g(u,v) = 2u+vThese two FUNCTION are now wisely CHAINED to build the FUNCTION G defined as follow :G(x,y) = g(u,v) = g( (u,v) ) = g(C(x,y)) = (goC)( (x,y) ) = (goC)(x,y)We now clearly see what is this FUNCTION G : THE CHAINED FUNCTION OF g and C :G = goCThe central "o" meaning the CHAINING (COMPOSITION) OPERATION ON FUNCTIONAL SPACE, defined, in general, for any two function A : E--> F, and B : F --> G, very simply as :BoA : E --> G , x --> BoA(x) = B(A(x))Which could thus, aternatively be named "THE BOA RULE" instead of the usual "fog" and "gof" of "FUNCTION CHAINING"... (joke!)In our exemple we then get, finaly, since u=x and v=y-x :G(x,y) = (goC)(x,y)  = g(C(x,y)) = g( (u,v) ) =  g(u,v) = g(x, y-x) = 2x + (y-x) = x +ySO WE "DISCOVER" THAT for all x and all y, we have :G(x,y) = f(x,y)That means FUNCTIONAL EQUALITY :G=fSo what have we done? Turning around? Yes, in a sens, because we have INSISTED in differentiating g with respect to x, IN A NUMERICAL SENS, instead of doing it in a FUNCTIONAL PROPER WAY that is, differentiating with respect to its FIRST variable in the ALGORITHM g : (@,#) --> g(@,#) = 2@ + #Thus, in this NUMERICAL SENS, x is no longer a DUMMY VARIABLE, but a "NUMBER ATTACHED TO f"! The physicist would call it a "physical observable". This way of looking at things makes it complicate and cand lead to confusions if one doesn't clearly understand the CHAIN FUNCTION KEY AND THE IN FACT THE MANIFOLD CHARTS MAPPING (THUS CHANGE OF CHARTS TRANSFORMATIONS, often differentiable for a regular manifold) laying under all this...  We thus have, in a sens, "turned around", leading to G=f! But doing so we nevertheless HAVE CLEARLY WRITEN THINGS IN CHAIN FUNCTION PROPER WAY, that reveal to be EXTREMELY USEFULL in higher and more abstract DIFFERENTIAL GEOMETRY, where crucial concepts of COVERING and UNIVERSAL COVERING, TANGENT SPACE, GROUP, LIE ALGEBRA, LIE DERIVATIVE, COVARIANT DERIVATIVE, CONNEXIONS, DIFFERENTIAL FORMS,..for instance, are usefully springing, needing CHAINING FUNCTION PROCESS to be clearly understood and mastered!Finally, often, this abuse of notation which misleadingly sets f=g, will be handled without damage by the "human computer", with some habits and agility. BUT only in rather easy area. On the contrary it will lead to catastrophic confusions, misconceptions, darmatically false results and chear non sens interpretations, in more suttle area...alike, not for naming them, general relativity, riemanian differential geomedry, condensed matter, plasma, etc...CHAINING FUNCTION OPERATION is one of the most simple, but important one in all mathematics! It's the building block of nested structures, one inside the others alike Pandora box, BUT "BOXES" THAT REVEAL FASCINATING STRUCTURES ALIKE GROUP, ALGEBRA, CLIFFORD,...TOPOS... : PointsFunctions, from points to pointsFunctional (or Operator), from functions to functions, etc...All included IN...CATEGORY Groetendick SPIRIT...deap!####[]