Calculus gets easier when you discard all the unnecessary philosophical baggage. Differential is definitely a fraction while you're using it. And you can divide by zero if you don't look directly at the zero.
Man, I have an advanced degree in engineering heavy in math. But I didn't understand that at all. All I know is bprp is going to be up late tonight updating his " dy/dx isn't a fraction video" :-)
Differential forms (aka exterior calculus) it what comes after vector calculus. The latter works only in 3D whereas the former works in nD. You'll get a proper n-dimensional generalization of curl there. ...and the amazing generalized Stokes theorem - which is, in my humble opinion, the most beautiful theorem in all of math. ...even though I also don't really understand exterior calculus very well - but just enough to appreciate the magnificence of the theorem. All the classical vector calc integral theorems, some complex calc integral theorems, the fundamental theorem of calc - they all fall out as special cases of that one single (and simple!) theorem. ...well "simple" in the sense of being short to write down. The content is quite dense, though
Bachelor's is 4 years and isn't considered an advanced degree, Master's is 2 years plus a Bachelor's and is considered an advanced degree, Doctorate is 2 years plus a Master's and is an even more advanced degree. The number of years is a cultural memory idea and doesn't actually correspond to reality for a lot of people. Speaking as an American.
Leibnitz thought of dy/dx as the ratio of two actual (although infinitesimal) things. Abraham Robinson showed (1961, much later!) how you could add to the real numbers a value whose magnitude was less than the magnitude of any nonzero real and come up with something that was as consistent as real number arithmetic; another _model_ of the real numbers if you will. His approach provided the rigor that had been lacking in Leibnitz' intuitively appealing approach and turned out to be widely applicable to mathematical systems with the right (rather simple) order properties.
Still, even with Robinson's approach, the derivative is not really a fraction. Rather, it is the common standard part of a set of nonstandard fractions.
What I really dislike about the dy/dx notation is that it implies that the variable name x has a special meaning. That's not true. For example, the functions f, g: M -> N, defined by f(x)=x^2 and g(t)=t^2 are precisely the same. We need to choose a variable name to write down what the function does, but that choice is completely arbitrary and the variable name has, purely mathematically speaking, no meaning on its own. I will admit that the Leibniz notation is convenient and intuitive and that it can be very useful especially in natural sciences, but in my opinion it should be introduced to learners more carefully as it could easily lead to conceptual misunderstandings.
As a physicist, there is never a time when dy/dx is not a fraction, and there's nothing you math nerds can do about it ! :D JK, nice video, very clear, thanks. I would actually really like to dig deeper into this topic, but everytime I try the terseness of math textbooks stops me...
Don't overthink it, Mike. When math teachers say it's not a fraction, all they mean is that you shouldn't cancel the d's and do something like dy/dx = y/x. Once you learn more calculus, you'll know when it's ok to "abuse" notation and write things like dy = f'(x)dx. It's the same when your parents told you _"Don't play with fire"_ or your other math teacher told you _"Don't take the square root of negative numbers."_ What they really meant was: _"Don't do it just yet; wait till you know more."_
I don't get why it's not a fraction even in its "raw" form? It looks like a fraction, it behaves like a fraction, so what gives? Stuff like this works: dy/dx = f(x) then dx/dy = 1/f(x), all legit. And it's possible to "multiply" by dx in the separation of variables. Even the chain rule works, i.e. dy/dx = (dy/dz)(dz/dx). What's missing?
It's not a fraction since both dy and dx are, to be specific, not numbers but differentials. Those are functions in terms of variable x and for every given x they both kind "equal" to zero, so dy/dx, simply, is an undetermenent function of x (like (0*x^2)/(0*x)). if we look at the definition of a derivative dy/dx=delta(y)/delta(x) with delta(x)->0 we can see that delta(y)/delta(x) IS a rational function (a fraction), but its limit is not one.
I do not understand, that somebody learns, it wouldn't be a fraction. In germany/austria, you learn it, that it is a fraction. It is like Δy/Δx, while Δy/Δx = (y_2-y_1) / (x_2-x_1). If you choose x_2 ~ x_1, than you have dy/dx. In school (gymnasium), you have to explore the behaviour of a curve by "using a line" and there are two solutions, which students find: you can use a line, which goes through two lines of the curve (secant line) or you try to use a line with same the same slope in a point (tangent line). Than you calculate the slope by fraction. After that, you learn to use the notation of dy/dx instead of Δy/Δx and a little bit later the f'(x) notation.
Δy and Δx are numbers but dx and dy are not numbers. With differential forms you have have a ratio of differential forms like dy/dx that is kind of like a fraction but it still isn't one.
Δy and Δx are difference in y and x respectively, and therefore are numbers, but dx and dy are not numbers. And here in order to treat dy/dx kind of like a fraction, that "difference in x" is not the definition of dx. It is a differential form, which is not such a simple thing as a difference in two numbers.
It's good to see more differential forms content. Differential topology and geometry is the zenith of calculus and analysis. I always struggled to actually figure out what they meant by that. I guess they mean things like dy/dx+3/7=(7dy+3dx)/(7dx) maybe don't work, but if you treat them as their own separate "class", then some fractional rules are followed. dy/du du/dx=dy/dx (chain rule) d(ay)/d(bx)=a/b dy/dx (by chain rule and linearity of the exterior derivative) d(y+z)/dx=dy/dx+dz/dx (by linearity of the exterior derivative) 1/(dy/dx)=dx/dy (by chain rule) It's actually quite impressive how well it does that. Actually, I think they do work in that fractional example above, now that I'm thinking about it. (7dy+3dx)/(7dx) =(7dy)/(7dx)+(3dx)/(7dx) =d(7y)/d(7x)+d(3x)/d(7x) =7/7 dy/dx+3/7 dx/dx =dy/dx+3/7 All this just by the rules above, I'm fairly certain.
How (dy)/(dx) is a fraction. Differential Forms look at a setting with variables u,v. 0-forms: functions: f(u,v) 1-forms: f(u,v)du+g(u,v)du 2-forms: f(u,v)du^dv What are these “extra” parts du,dv: R^2
Loving this way of defining derivatives as fractions but too simple to work out if dx/dy for a spiked function f(y) is meaningful and whether there may be a grey area, that converges to your definition of a zero function, to treat spiked functions.
A more considered comment runs like this: but first! (dy/dx) (dx/dt) tends to dy/dt by cancelling the dx fractional parts. Sorta explains dx tending to zero nicely. Anyway... 6:08 - nicely done! I hope all listeners/viewers experience thoughts: tied variables? orhogonality? A nice thing about orthogonals is that a lot of messy stuff gets sent into null space no? 12:58 general observation: a robust or stout result in math should really stubbornly present itself however the result was approached upto isomorphisms? Therefore (dy/dx)(dx/dt) is isomorphic to dy/dt and if nicely so it is a 2 to 2 too
There are practically zero instances where it behaves otherwise, I almost forgot that it wasn't a fraction after all the time I have seen it acting like one.
Since the subject here is such a pedantic one, I feel compelled to suggest that by “fraction” you mean “ratio.” For one whose favorite function is the floor, you should know better 😉
It's not a fraction, in the sense that "normal" division is an operation performed on two actual real numbers, and dy and dx are not actual numbers. But the reason that the notation dy/dx is so widely used is that when we apply the laws of fractions to it, the result suggests mathematical processes that are still valid. "Separation of variables" actually works to solve some differential equations, for example. Pretending it's a real fraction and applying those laws, in other words, reflects intuitions that are mathematically true, so it's useful to continue pretending it's actually a fraction.
I'd love some more videos about differential forms. They're so weird to me. There's just something about them that hasn't clicked yet despite my attempts
I am not mathematician so I still don't fully get what they mean by it's not a fraction but it's always a bit weird for me considering that the definition of a derivative is the limit of a fraction and that the chain rule follows the same kind of rules fractions do And it also makes intuitive sense since a derivate is the proportion of the variation of two variables I am guessing that maybe for some very specific things the fraction rules don't quite apply?
what I never understood is what exactly is dv/du in differential forms terms? Is it just a function that takes in a single vector? How exactly do you evaluate it?
Is it not a ratio??? For ratios can be represented as a fraction as well. Still this video confuses me more than it clears my doubts of Calc 1 now that the video is titled in such a way
@@Vega1447 "he chose u =(a\\b) and v=(c\\d)" Err - no?!? Where in the video do you think he did do that?!? He explains what happens when you _apply_ the differential forms, i. e. the _functions_ du and dv, to the vectors (a,b) and (c,d). Nowhere does he say that u = (a,b) and v = (c,d)! Where did you get that from?!
Well, it would have been true if you were able to actually divide one form with respect to another, but you can’t do that xD I would rather have seen an intro in the non-standard calculus with an extension of the set of reals to include infinitesimals, where you could make this into a fraction :) But what was shown in the video, if you ever do this on the exam, you will be disappointed in the result xD
You can define division easily. Let dy/dx equal some function that, if multiplied by dx, is equal to dy. (If such function doesn't exist then the operation is undefined). Exactly how division within the real numbers can be defined.
@@coc235 while I understand the intuition behind your proposition, in my opinion, it is, nonetheless, ill-formed. dx and dy in your example are 1-forms, meaning that they take vector fields to numbers. The proposed ratio is a function that takes points in on the manifold to numbers. This is a stretch that I cannot accept. In my book, if you want to define division of something by something, you at least need to preserve the domains of the functions in play! In your example, however, the domain of two operands is a space of vector fields, and the domain of the resulting function is a manifold itself. Dont like it, sorry :) Unless of course it is an entire point of your division operator: take a pair of forms and return a function... Well, maybe, idk, feels iffy. still, dx needs to be nowhere vanishing at least, to add to your definition :)
You haven’t actually proved anything. You are simply manipulating notation. In Newtonian calculus, symbol, d/dxf(x) is short hand for taking a limit of difference quotients of the value of a function, f, at a particular point, x, which may or may not exist. It can also be thought of as an operator from one function space to another. The symbols, du and dv, are essentially differentials which represent the variation of the value of a function at x due to small changes. You’re basically back at Berkeley and Cauchy
@ Yes, Leibniz introduced that notation. I use the term, “Newtonian”, to differentiate it from more general calculus such as the calculus of variation. It would have been more accurate to say, “Newtonian/Leibniz calculus”. But that doesn’t change my comment
"You haven’t actually proved anything." The point of the video was not to "prove" anything. "The symbols, du and dv, are essentially differentials which represent the variation of the value of a function at x due to small changes." That's true for Leibniz calculus. But it's not true in this video, where du and dv are differential forms - as he pointed out several times. "You’re basically back at Berkeley and Cauchy" Not at all. Thanks for showing that you haven't understood the basics of differential forms.
@@bjornfeuerbacher5514 Why the hostility? He claimed that dy/dx was a true fraction. However, neither the “numerator” or the “denominator” are numbers (division is really an algebraic property of real numbers). The symbol represents df(x)/dx, where f(x)=y. In other words, the symbol, “d/dx” represents lim((f(x+h)-f(x))/h) as h-> 0, not the simple division of two numbers. Also, du and dv are not differential forms. They are components of 1- and 2-forms. Check his definition in the video. He is merely confusing the symbol, “d” which is used in both the Leibniz and the differential forms notations to prove his claim. However, they have different meanings.
Now that's a good thumbnail. As a physicist I clicked immediately 😂
Same here! XD
Same…
How to attract physicist fr fr
Reminds me of the joke
Mathematicians: dy/dx is not a fraction!
LaTeX: \frac{dy}{dx}
Mathematicians: 😮
\frac{\mathrm{d}y}{\mathrm{d}x}
Calculus gets easier when you discard all the unnecessary philosophical baggage. Differential is definitely a fraction while you're using it. And you can divide by zero if you don't look directly at the zero.
Man, I have an advanced degree in engineering heavy in math. But I didn't understand that at all. All I know is bprp is going to be up late tonight updating his " dy/dx isn't a fraction video" :-)
What do you by an 'advanced degree' ? In France we don't have that concept so I'm curious
Tensors are used in some engineering branches, but this video is more about tensor calculus/differential geometry stuff.
Differential forms (aka exterior calculus) it what comes after vector calculus. The latter works only in 3D whereas the former works in nD. You'll get a proper n-dimensional generalization of curl there. ...and the amazing generalized Stokes theorem - which is, in my humble opinion, the most beautiful theorem in all of math. ...even though I also don't really understand exterior calculus very well - but just enough to appreciate the magnificence of the theorem. All the classical vector calc integral theorems, some complex calc integral theorems, the fundamental theorem of calc - they all fall out as special cases of that one single (and simple!) theorem. ...well "simple" in the sense of being short to write down. The content is quite dense, though
Bachelor's is 4 years and isn't considered an advanced degree, Master's is 2 years plus a Bachelor's and is considered an advanced degree, Doctorate is 2 years plus a Master's and is an even more advanced degree. The number of years is a cultural memory idea and doesn't actually correspond to reality for a lot of people. Speaking as an American.
@@jbragg33 Master's degree. Between bachelor's and PhD. Heavy in numerical methods and vector calculus for my thesis.
What allows the division at 15:30? Without a course in abstract algebra, this is pretty much like saying "dy/dx is a fraction because I say so"
It's a fraction of infinitesimals.
Leibnitz thought of dy/dx as the ratio of two actual (although infinitesimal) things. Abraham Robinson showed (1961, much later!) how you could add to the real numbers a value whose magnitude was less than the magnitude of any nonzero real and come up with something that was as consistent as real number arithmetic; another _model_ of the real numbers if you will. His approach provided the rigor that had been lacking in Leibnitz' intuitively appealing approach and turned out to be widely applicable to mathematical systems with the right (rather simple) order properties.
sometimes i feel we are still struggling with the axiomatization of analysis
Still, even with Robinson's approach, the derivative is not really a fraction. Rather, it is the common standard part of a set of nonstandard fractions.
What I really dislike about the dy/dx notation is that it implies that the variable name x has a special meaning. That's not true. For example, the functions f, g: M -> N, defined by f(x)=x^2 and g(t)=t^2 are precisely the same. We need to choose a variable name to write down what the function does, but that choice is completely arbitrary and the variable name has, purely mathematically speaking, no meaning on its own. I will admit that the Leibniz notation is convenient and intuitive and that it can be very useful especially in natural sciences, but in my opinion it should be introduced to learners more carefully as it could easily lead to conceptual misunderstandings.
As a physicist, there is never a time when dy/dx is not a fraction, and there's nothing you math nerds can do about it ! :D JK, nice video, very clear, thanks. I would actually really like to dig deeper into this topic, but everytime I try the terseness of math textbooks stops me...
Don't overthink it, Mike. When math teachers say it's not a fraction, all they mean is that you shouldn't cancel the d's and do something like dy/dx = y/x. Once you learn more calculus, you'll know when it's ok to "abuse" notation and write things like dy = f'(x)dx.
It's the same when your parents told you _"Don't play with fire"_ or your other math teacher told you _"Don't take the square root of negative numbers."_ What they really meant was: _"Don't do it just yet; wait till you know more."_
I don't get why it's not a fraction even in its "raw" form? It looks like a fraction, it behaves like a fraction, so what gives? Stuff like this works: dy/dx = f(x) then dx/dy = 1/f(x), all legit. And it's possible to "multiply" by dx in the separation of variables. Even the chain rule works, i.e. dy/dx = (dy/dz)(dz/dx). What's missing?
It's not a fraction since both dy and dx are, to be specific, not numbers but differentials. Those are functions in terms of variable x and for every given x they both kind "equal" to zero, so dy/dx, simply, is an undetermenent function of x (like (0*x^2)/(0*x)).
if we look at the definition of a derivative dy/dx=delta(y)/delta(x) with delta(x)->0
we can see that delta(y)/delta(x) IS a rational function (a fraction), but its limit is not one.
Isn't it Lagrange notation that uses the prime-symbol. Newton used dots... are those also called 'prime symbols'?
I think you meant Leibniz notation
@@jbragg33 No, he is right. Leibniz used the notation dy/dx. Newton used dots. Lagrange introduced the prime.
^^^^^
@@bjornfeuerbacher5514 Newton called it the flux or fluxion.
Unless you learned from an old school teacher.
I do not understand, that somebody learns, it wouldn't be a fraction. In germany/austria, you learn it, that it is a fraction. It is like Δy/Δx, while Δy/Δx = (y_2-y_1) / (x_2-x_1). If you choose x_2 ~ x_1, than you have dy/dx.
In school (gymnasium), you have to explore the behaviour of a curve by "using a line" and there are two solutions, which students find: you can use a line, which goes through two lines of the curve (secant line) or you try to use a line with same the same slope in a point (tangent line). Than you calculate the slope by fraction. After that, you learn to use the notation of dy/dx instead of Δy/Δx and a little bit later the f'(x) notation.
Δy and Δx are numbers but dx and dy are not numbers. With differential forms you have have a ratio of differential forms like dy/dx that is kind of like a fraction but it still isn't one.
its obviously a fraction --- its the difference of y divided by the difference of x
(y sub 2 minus y sub 1) divided by (x sub 2 minus x sub 1)
Δy and Δx are difference in y and x respectively, and therefore are numbers, but dx and dy are not numbers. And here in order to treat dy/dx kind of like a fraction, that "difference in x" is not the definition of dx. It is a differential form, which is not such a simple thing as a difference in two numbers.
Your forgot taking the limit, i.e., lim delta y/delta x
It's good to see more differential forms content. Differential topology and geometry is the zenith of calculus and analysis.
I always struggled to actually figure out what they meant by that. I guess they mean things like
dy/dx+3/7=(7dy+3dx)/(7dx)
maybe don't work, but if you treat them as their own separate "class", then some fractional rules are followed.
dy/du du/dx=dy/dx (chain rule)
d(ay)/d(bx)=a/b dy/dx (by chain rule and linearity of the exterior derivative)
d(y+z)/dx=dy/dx+dz/dx (by linearity of the exterior derivative)
1/(dy/dx)=dx/dy (by chain rule)
It's actually quite impressive how well it does that.
Actually, I think they do work in that fractional example above, now that I'm thinking about it.
(7dy+3dx)/(7dx)
=(7dy)/(7dx)+(3dx)/(7dx)
=d(7y)/d(7x)+d(3x)/d(7x)
=7/7 dy/dx+3/7 dx/dx
=dy/dx+3/7
All this just by the rules above, I'm fairly certain.
How (dy)/(dx) is a fraction. Differential Forms look at a setting with variables u,v. 0-forms: functions: f(u,v) 1-forms: f(u,v)du+g(u,v)du 2-forms: f(u,v)du^dv What are these “extra” parts du,dv: R^2
Loving this way of defining derivatives as fractions but too simple to work out if dx/dy for a spiked function f(y) is meaningful and whether there may be a grey area, that converges to your definition of a zero function, to treat spiked functions.
I'm so Stokes d.
When splitting hairs, a differential form is NOT a derivative. It is shorthand form of writing "delta forms", and it gives the same results.
18:24
A more considered comment runs like this: but first! (dy/dx) (dx/dt) tends to dy/dt by cancelling the dx fractional parts. Sorta explains dx tending to zero nicely. Anyway...
6:08 - nicely done! I hope all listeners/viewers experience thoughts: tied variables? orhogonality?
A nice thing about orthogonals is that a lot of messy stuff gets sent into null space no?
12:58 general observation: a robust or stout result in math should really stubbornly present itself however the result was approached upto isomorphisms?
Therefore (dy/dx)(dx/dt) is isomorphic to dy/dt and if nicely so it is a 2 to 2 too
Getting my bachelor’s in mechanical engineering, we treated it like a fraction all of the time.
There are practically zero instances where it behaves otherwise, I almost forgot that it wasn't a fraction after all the time I have seen it acting like one.
Since the subject here is such a pedantic one, I feel compelled to suggest that by “fraction” you mean “ratio.” For one whose favorite function is the floor, you should know better 😉
It's not a fraction, in the sense that "normal" division is an operation performed on two actual real numbers, and dy and dx are not actual numbers. But the reason that the notation dy/dx is so widely used is that when we apply the laws of fractions to it, the result suggests mathematical processes that are still valid. "Separation of variables" actually works to solve some differential equations, for example.
Pretending it's a real fraction and applying those laws, in other words, reflects intuitions that are mathematically true, so it's useful to continue pretending it's actually a fraction.
So, dy/dx walks like a duck?
Wandering explanation...... congratulations 🎉 sir ...
I'd love some more videos about differential forms. They're so weird to me. There's just something about them that hasn't clicked yet despite my attempts
I am not mathematician so I still don't fully get what they mean by it's not a fraction but it's always a bit weird for me considering that the definition of a derivative is the limit of a fraction and that the chain rule follows the same kind of rules fractions do
And it also makes intuitive sense since a derivate is the proportion of the variation of two variables
I am guessing that maybe for some very specific things the fraction rules don't quite apply?
what I never understood is what exactly is dv/du in differential forms terms? Is it just a function that takes in a single vector? How exactly do you evaluate it?
Is it not a ratio??? For ratios can be represented as a fraction as well. Still this video confuses me more than it clears my doubts of Calc 1 now that the video is titled in such a way
bruh i thought it was going to be about nonstandard analysis :c
dy/dx is both a fraction and not a fraction.
See my(Robert Telarket) presentation paragraph 11 in Quora of What is a derivative in layman's terms.
Finally!
I'm puzzled. If u and v are vectors in \R^2, what is the partial derivative of f(x,y) wrt x or y? Partials are wrt a scalar variable.
x and y are not scalars, they are components of a vector. And where did he say that u and v are vectors in R² ?!
@bjornfeuerbacher5514 he chose u =(a\\b) and v=(c\\d)
Typo I should have said partial derivatives wrt u and v.
@@Vega1447 "he chose u =(a\\b) and v=(c\\d)"
Err - no?!? Where in the video do you think he did do that?!? He explains what happens when you _apply_ the differential forms, i. e. the _functions_ du and dv, to the vectors (a,b) and (c,d). Nowhere does he say that u = (a,b) and v = (c,d)! Where did you get that from?!
Well, it would have been true if you were able to actually divide one form with respect to another, but you can’t do that xD I would rather have seen an intro in the non-standard calculus with an extension of the set of reals to include infinitesimals, where you could make this into a fraction :) But what was shown in the video, if you ever do this on the exam, you will be disappointed in the result xD
You can define division easily. Let dy/dx equal some function that, if multiplied by dx, is equal to dy. (If such function doesn't exist then the operation is undefined). Exactly how division within the real numbers can be defined.
@@coc235 while I understand the intuition behind your proposition, in my opinion, it is, nonetheless, ill-formed. dx and dy in your example are 1-forms, meaning that they take vector fields to numbers. The proposed ratio is a function that takes points in on the manifold to numbers. This is a stretch that I cannot accept. In my book, if you want to define division of something by something, you at least need to preserve the domains of the functions in play! In your example, however, the domain of two operands is a space of vector fields, and the domain of the resulting function is a manifold itself. Dont like it, sorry :) Unless of course it is an entire point of your division operator: take a pair of forms and return a function... Well, maybe, idk, feels iffy. still, dx needs to be nowhere vanishing at least, to add to your definition :)
So are you a liar Mike?
Mike, ur getting worse and WORSE on ur explanations (sorry)
Then why don't you make a better one quizkid ?
You haven’t actually proved anything. You are simply manipulating notation. In Newtonian calculus, symbol, d/dxf(x) is short hand for taking a limit of difference quotients of the value of a function, f, at a particular point, x, which may or may not exist. It can also be thought of as an operator from one function space to another. The symbols, du and dv, are essentially differentials which represent the variation of the value of a function at x due to small changes. You’re basically back at Berkeley and Cauchy
"in Newtonian calculus, symbol, d/dxf(x) is short hand"
That's not Newtonian calculus, that's Leibniz' notation.
@ Yes, Leibniz introduced that notation. I use the term, “Newtonian”, to differentiate it from more general calculus such as the calculus of variation. It would have been more accurate to say, “Newtonian/Leibniz calculus”. But that doesn’t change my comment
"You haven’t actually proved anything."
The point of the video was not to "prove" anything.
"The symbols, du and dv, are essentially differentials which represent the variation of the value of a function at x due to small changes."
That's true for Leibniz calculus. But it's not true in this video, where du and dv are differential forms - as he pointed out several times.
"You’re basically back at Berkeley and Cauchy"
Not at all. Thanks for showing that you haven't understood the basics of differential forms.
@@bjornfeuerbacher5514 Why the hostility? He claimed that dy/dx was a true fraction. However, neither the “numerator” or the “denominator” are numbers (division is really an algebraic property of real numbers). The symbol represents df(x)/dx, where f(x)=y. In other words, the symbol, “d/dx” represents lim((f(x+h)-f(x))/h) as h-> 0, not the simple division of two numbers. Also, du and dv are not differential forms. They are components of 1- and 2-forms. Check his definition in the video. He is merely confusing the symbol, “d” which is used in both the Leibniz and the differential forms notations to prove his claim. However, they have different meanings.
I thought this video would go the non standard analysis route with infinitesimals
Isn't it "like" a ratio of infinitesimals?