Algebraic Operations on Differentials in Liebniz Notation: An Abuse?

[IniquiTrance]

Why is it that performing algebraic operations on differentials in Liebniz notation is considered an abuse of the notation?

In every case, it seems like the operation makes sense to me.

Chain rule etc:

\frac{dy}{du}\frac{du}{dx} = \frac{dy}{dx}

Since differentials are just limits, if they exist, then they're regular fractions.

Thanks!

 

[slider142]

Since differentials are just limits, if they exist, then they're regular fractions.

No, they are not. Depending on the actual terms in the limit, the behavior of a limit may be quite different from the algebraic form of the term being considered. Consider the example in this post, which a student may mistakenly believe to be correct analysis inspired by the ease of Leibnitz notation. Do not trust the notation too much; always check the analysis.

 

[chrisr999]

Derivatives ARE fractions.
By definition, a derivative is the slope or gradient of the tangent to the graph. The tangent is a straight line and the limit is necessary to find the gradient of the tangent rather than a line cutting through two points on the curve in the graph!
How we combine derivatives is as follows...
suppose we want the derivative of {(2x+3) squared} with respect to x. We simply multiply the derivative of {2x+3 squared} with respect to 2x+3 by the derivative of 2x+3 with respect to x,
because it is the same as the derivative of {2x+3 squared} with respect to x.
We are "multiplying by 1".
It can be performed within "first principles notation" or the shorter notation of Leibniz and co.

 

[g_edgar]

When you try the same thing for changing variables in double integrals you will be in trouble. To change from dx dy to du dv you have to multiply by the Jacobian.

P.S. Leibniz not Liebniz

 

[chrisr999]

of course, that is integration, the calculation of area, volume etc,
in the forum question, we simply multiply by 1 in the form of dy/dy, dq/dq and so on and swop around the denominators as in this situation (gradient slopes), we are undoubtedly dealing with fractions (ie 10/3 times 6/5 is 10/5 times 6/3 equals 4 as both are 60/15). That is the simple technique where that technique is applicable!

 

[chrisr999]

Slider,
i disagree with your analysis.
If x+y+z=0 then x=-(y+z), y=-(x+z) and z=-(x+y).

dy/dx is -1-dz/dx
dz/dx is -1-dy/dx
dx/dz is -1-dy/dz
dy/dz is -1-dx/dz
dx/dy is -1-dz/dy etc

and so on, since if any of x or y or z change, the others must change such that they still sum to zero and we must rule out the possibility of one of the three being a constant, which has not been specified by the equation, though you say they are all variables but the nature of their variation must be specified. You cannot state the derivatives the way you have done as you are suggesting x does not affect y, z does not affect x and y does not affect z which is contradictory

 

[jgens]

Slider,
i disagree with your analysis.
If x+y+z=0 then x=-(y+z), y=-(x+z) and z=-(x+y).

dy/dx is -1-dz/dx
dz/dx is -1-dy/dx
dx/dz is -1-dy/dz
dy/dz is -1-dx/dz
dx/dy is -1-dz/dy etc

and so on, since if any of x or y or z change, the others must change such that they still sum to zero and we must rule out the possibility of one of the three being a constant, which has not been specified by the equation, though you say they are all variables but the nature of their variation must be specified. You cannot state the derivatives the way you have done as you are suggesting x does not affect y, z does not affect x and y does not affect z which is contradictory

You're clearly unaware of partial derivatives, especially since slider's analysis (Hurkyl's actually) is entirely correct. Derivatives are not quotients, they are limits of quotients. Without the limit concept, derivatives are useless fractions of the form 0/0. Because of this fact that, we cannot treat them as fractions in any rigorous sense.

 

[snipez90]

Well if you are just taking derivatives, I don't really see a problem. I don't think anyone would compute a derivative incorrectly because he did not understand the notational peculiarities.

Now if you were to prove anything, an appeal to Leibniz notation is probably not a good idea. This is pretty clear I think. As for Leibniz notation itself, I think the abuse of notation is exaggerated. Again, computationally, I doubt many people would be lead astray by intuition, as long as they realize that it is pretty hard to be rigorous about infinitesimally small quantities.

But still, the OP probably made the most important observation regarding the continued use of Leibniz notation. Leibniz notation is popular because it is flexible and useful. The operations should make sense because Leibniz notation allows us to treat dy and dx in exactly the same way as if they were ordinary numbers. This is why it is so handy in calculations. But whatever, taking a derivative is kind of boring anyways.

 

[chrisr999]

the context assumed is "derivative as the tangent gradient".

0/0 is not a fraction! zero is not a number!
When calculating a derivative, we must find a way to "cancel" the infinitesimally small quantity in the denominator, typically for instance by locating it as a factor in the numerator.
As an example, what is {x(squared)-9}/{x-3} when x=3?
It is the fraction X+3 written in a deceptive way.
The trouble with 0/0 is this...
How many zeroes is the numerator and how many zeroes is the denominator?
since 0(1)=0(2)=0(3)=0(4), keep doing nothing and end up with nothing.
It's undefined because we have lost the details of how many times the denominator is the numerator.
Even though derivatives are calculated with limits, since we are dealing with tangents to a single point, not a line through two points... if the limit exists the derivative is a line gradient which is a quotient, no matter how we refer to it.
It works where it is applicable and not where it is not applicable. If the partial derivatives are across different unrelated variables it won't work!
We cannot divide miles by hours and give our answer in km/hour without converting correctly first. Slider's analysis is incomplete. he needs to state more clearly the relationship between x, y and z.
When performing partial differentiation, you are holding one of these constant. This is different to finding the derivative of a function of a function (say x).
Mathematics has many areas that seem the same but we may need to be more specific.
For instance CosA is either the ratio of two sides in a right angled triangle or the x co-ordinate of a point on the circumference of a unit radius circle, computed using the right-angled triangle.
In the context of the Chain Rule of differentiation, we are only "multiplying by 1" in a way that suits us, similarly to how we combine fractions with different denominators. If the word "derivative" means something else in a slightly different context, we are playing a different game of football.
When calculating the volume of a cube, the variables are "unrelated". Width, height and depth are "independent" unless restrictions are placed on the volume. Under these circumstances dx, dy and dz may not have any interrelationship as they are infinitesimal lengths independent of each other.
Partial derivatives are "partial", used in situations where the function depends on more than one "independent" variables. it's a different ball game. For the variables that are independent, there is no derivative as we cannot draw of a graph of one versus the other.
In Slider's analysis he obtains three partial derivatives multiplied together to get -1 when we'd expect 1 if we were dealing with derivatives. This is because he is calculating partial derivatives, not derivatives, by holding a variable constant.
A partial derivative is not a complete derivative. It is a slight redefinition of a derivative to deal with multivariable situations, a variation of a context.

 

[HallsofIvy]

And, let me point out, it is quite possible for a function of several variables to have partial derivatives at a point where it is NOT differentiable.

 

[chrisr999]

It is true that we cannot treat derivatives as fractions in cases of multiple variables such that (dy/dx)(dx/dz)(dz/dy) is not equal to one, in other words, the "numerators" and "denominators" do not cancel as fractions normally do, but this is due to being in a three-dimensional space.
In the situation where we have a function of a function of x, the Chain Rule extracts the derivative of a composite function by multiplying by 1 and juggling the denominators.
In the case of multiple variables however, the partial derivatives are fractions but "not as we know them"!
In other words, they are not "combinable as fractions" until we clear up what kind of fractions they are, similar to the 0/0 case which has hidden the details of the fraction.
In the partial derivatives case, since we have not simply multiplied dy/dx by du/du to get (dy/du)(du/dx) by swopping the denominators, allowing us to differentiate this type of "composite function", we do not have the liberty of swopping around the numerators and denominators. Our context has changed.
To treat the notation as fractions, the context needs to be checked to see if the numerator symbol is really equal to the denominator symbol in another fraction!
I think the thread was started assuming functions of a single variable, however the introduction of the partial derivatives shows the importance of knowing the context.

 

[Hurkyl]

Derivatives ARE fractions.

No, they're not. (First derivatives in one-dimensional parameter spaces) are merely similar to fractions. e.g.

multiplied dy/dx by du/du to get (dy/du)(du/dx) by swopping the denominators

you have this backwards. We don't swap denominators because derivatives are fractions -- we say derivatives are similar to fractions because we've proven by other means that you can "swap" denominators.

One of the dangers of lying to students and telling them derivatives are fractions is that they might believe you -- and they'll start doing all those other things you can do with fractions that don't work here.

e.g. they'll immediately start to wonder what dx all by itself would mean. If they figure out the idea of a differential form... wonderful! But if they don't and get some other idea stuck in their head because we pointed them down that road and gave them no guidance... bad.

Or another example: if derivatives are fractions, then obviously the following should be an identity, right?
\frac{d^2 f}{(dy)^2} \cdot \left( \frac{dy}{dx} \right)^2 = \frac{d^2 f}{(dx)^2}

There are wonderful concepts available in Calc I for the students to learn -- e.g. things like the idea of a first-order approximation. Pretending we're just doing fancy things with fractions would (appear to) obstruct their chances of learning that.

 

[chrisr999]

it would pay to clear up the language.
A derivative is a fraction because it mathematically calculates the gradient of a tangent to a curve, that's a fraction since a line gradient is (delta y)/(delta x).
however, if you are operating with different variables, the fractions may not be compatible!
that's clear,
derivatives work the same way as fractions when you have compatibility. don't forget dy/dx and so on is a notational short cut and is really tan(angle) for the tangent.

derivatives are fractions because a line gradient is.
a derivative is a line gradient, no two ways about it
BUT there are situations when you can use the fraction math because we are dealing with compatible fractions. there are obvious cases where you can't.
If students learn what is actually being calculated, they can learn without ambiguity.
Telling them derivatives are not fractions is only adding to the confusion.
There is far more to the picture.

 

[chrisr999]

That is being way too obvious, Hurkyl,
of course that situation is not a pair of fractions where you can simply cancel notational terms. A dialogue should clear up confusion.
One of the difficulties with calculus is the notation.
It was developed from sound principles using straight lines, whereby a gradient has a numerator and denominator, leading to a value evaluated at a limit as the denominator approaches zero.
In certain situations, when dealing with a single variable, the "quotient" can be dealt with as a fraction if you work it through in first principles form. however you cannot say in general that we can simply cancel "like symbols" such as dy, dx, dz etc.
It all depends on the situation being analysed.
You cannot exclusively say that derivatives are not fractions, nor can you exclusively say that a dx in a numerator position must cancel with a dx in a denominator position.
They are much more subtle fractions.
It would pay a student to understand the different cases.

Consider the graph of y=x(squared), or similar (single variable).
if you draw the graph and pick a point x,y on the graph, then if you draw the tangent, you can view dy/dx or d(f[x])/dx using delta(y)/delta(x) for the tangent itself.
if you turn the page of the graph to it's side, you can now view dx/dy or dx/d(f[x]), which is the gradient to x = f(y) at the same point (x,y). One of these is a/b and the other is b/a.
In situations of a single variable like this (dy/dx)(dx/dy) is 1 as we have (a/b)(b/a).

It is not that dx is the numerator and dy the denominator!
That is notation! written like a fraction for a reason.
The ratio of dy to dx however is tan(angle) for the tangent.

 

[Hurkyl]

That is being way too obvious, Hurky,
of course that situation is not a pair of fractions where you can simply cancel notational terms.

You say "of course", but it's not going to be obvious to the student -- especially if you're telling him derivatives are fractions. (and students aren't the only people who make this mistake!)

A dialogue should clear up confusion.

Yes, that's right. And the dialongue should emphasize to the student that derivatives aren't fractions, despite having some similar arithmetic properties.

If you want to redefine the word "fraction" to include both fractions and derivatives, that's fine -- but you need to make sure the student understands that you've done that. (aside: I really don't see why using the same word for both concepts should be preferable)

 

[chrisr999]

Students learn by reference to previous learning.
Then the student can say "this is like that but different in this new way",
by referring to previous knowledge. Otherwise, Mathematics becomes disjointed, learning becomes disjointed and when that happens, we have the ground for confusion.
Mathematics is a universal language, though we may have various ways of wording it.
Understanding the similarities and dissimilarities generates comprehension, breakthroughs in understanding.
I've seen a student totally bewildered by the statement in a textbook that "derivatives are not fractions" and began to struggle with the Chain Rule.
They may not be constant/constant due to the changing direction of the graph but derivatives are tangent gradients.
If we start saying derivatives are not fractions, then we should clear up what we mean, we are certainly not saying a derivative is anything other than the gradient of a tangent, and the gradient of a tangent is delta(y)/delta(x). We cannot choose two points on the curve to find this but we can choose any two points on the tangent.
Students first learn with averages in school. The speeds they work with are average speeds, then we introduce calculus to show how we calculate instantaneous speed by letting the time interval approach zero, graphically speaking we are bending a line until it is a tangent to the graph of distance versus time. By finding the limit (as delta t goes to zero) we are discovering the gradient of the tangent, which of itself is a quotient representing distance/time at an instant in time.

 

[chrisr999]

Although we can call dy/dx "notation",
it's really tan(angle) of the tangent to the curve. It may be impossible to see at the limit, but if you magnify it greatly, then it is the right-angled triangle flush against the tangent, with the perpendicular sides parallel to both axes.
That's what we mean by calling it a fraction.
dy/dx is delta(y)/delta(x) for the tangent, or opposite/adjacent, since proportions are maintained at any scale for equiangular triangles.

dy/dx for the curve is delta(y)/delta(x) for the tangent. For a single variable, y=f(x) then for a point (x,y) derivatives on either axis can be combined as fractions.

 

[Hurkyl]

Students haven't learned about differential forms or number systems with nonzero infinitessimals yet or whatever technical idea is needed to make sense of your exposition. Your attempts to "explain" the derivative is not "referencing previous learning". Instead, you are introducing new ideas and coercing the student to think in terms of the new ideas, despite the fact you are going to give them absolutely zero guidance in the understanding of those ideas.


Yes, there is previous learning to reference. One has the idea of tangent lines -- and (AFAIK) the first thing most textbooks do to introduce derivatives is the heuristic method of computing the tangent line as the limiting secant line.

(I put limit in bold because it's really important, but it's something you appear to be downplaying)

One also knows of inequations, in terms of which the important idea of differential approximation can be introduced. (I assert that differential approximation is by far the most important concept in Calc I)

There's also the idea of "rate of change", but the more I think about it, the more I think that really ought to be delayed until after the fundamental theorem of calculus, so that the student can be shown how the "rate of change" relates to actual changes.

 

[OrderOfThings]

Or another example: if derivatives are fractions, then obviously the following should be an identity, right?
\frac{d^2 f}{(dy)^2} \cdot \left( \frac{dy}{dx} \right)^2 = \frac{d^2 f}{(dx)^2}

This is a correct identity. The fractions d^2 f/dx^2 and d^2 f/dy^2 equal the second derivative of f only under the assumption that d^2x and d^2y vanish. The general formula for the second derivative of f with respect to x is
\frac{d^2\!f\,dx-df\,d^2x}{dx^3}

 

[chrisr999]

I might post a diagram to explain what I was trying to explain Hurkyl,
the person who started the post wanted to understand how derivatives relate to ratios.
That can easily be shown using a tangent at a specific point and showing that dy/dx is a regular ratio at that point and the value of the ratio varies as you move around the graph.
At a specific point derivatives can be worked as fractions.
a clear geometric analysis can show this.
If you want to help that's ok but derivatives are pretty much geometrical creatures and the notation has to be developed around that

 

[chrisr999]

I've attached a portable document file I developed today, explaining the situation in which derivatives can be treated as regular fractions and their nature in general for one variable functions.
Hope this helps,
(chris)

 

[jgens]

This is what I'm struggling with:

"The numerator is not "dy" and the denominator is not "dx" in the accepted sense of the fraction, since what matters is the "ratio" of "dy" to "dx"."

Unless you introduce a number system with infinitessimals (not the real numbers), how can you make sense of a statement like "ratio of dy to dx"?

 

[chrisr999]

You see we can write a number in an infinite number of ways.

1/2 means the denominator is twice the numerator and we can write it as 1/2, 10,20, 720/1440, 0.00000035/0.0000007 and so on.

Or 5 is 20/4 or 1500/300 or 0.000001/0.0000005.

A value is determined by the "ratio" of the numerator and denominator, not their specific values. One value is "so many times" the other.

Of course, to treat derivatives as having constant values that can cancel when combining fractions together, we must be at the same position on a graph for those derivatives that we work as fractions with the value of numerator and denominator already determined,
as derivatives are fractions that vary with the value or position of the variable.

Remember dy and dx may be infinitesimally small but they form a ratio and we cannot simply say the derivative is an "unchanging" fraction.

For a fixed fraction, the numerator and denominator can be written an infinite number of ways but the "ratio" never changes.

That ratio changes for non-linear functions as the rate of change of the function depends on the value of the function variable itself, or the position on the graph at which the derivative is calculated.

For instance (dy/dx)(dx/dy) will not be unity if these two derivatives were calculated at different values of x. If they are, it's either coincidental or we chose 2 points a cycle apart on a periodic function.

all of this is a verbal description of the geometry of derivatives

 

[Hurkyl]

This is a correct identity. The fractions d^2 f/dx^2 and d^2 f/dy^2 equal the second derivative of f only under the assumption that d^2x and d^2y vanish. The general formula for the second derivative of f with respect to x is
\frac{d^2\!f\,dx-df\,d^2x}{dx^3}

I'm curious what you think "d^2 f/dx^2" means (along with "d^2 x") -- when I learned calculus, that literally meant "the second derivative of f with respect to x then x"

 

[jgens]

Remember dy and dx may be infinitesimally small but they form a ratio and we cannot simply say the derivative is an "unchanging" fraction.

I understand that if we introduce infinitessimals that they form a ratio. However, if we work strictly within the real number system (this is all most calculus students are familiar with when they learn the subject) a ratio of dy to dx makes no sense because infinitessimals do not exist.

 

[Hurkyl]

explaining the situation in which derivatives can be treated as regular fractions and their nature in general for one variable functions.
Hope this helps,
(chris)

This isn't anything new -- this suffers exactly from the criticism I made in post #18. You're introducing infinitessimals, but
(1) aren't treating them consistently
(2) don't teach the student about them
See Elementary Calculus: An Infinitessimal Approach by Keisler for an example of how to go about teaching calculus via infinitessimals.

 

[chrisr999]

To that last comment, i don't know what to say!
I guess my average score of 100 per cent in higher level mathematics means nothing to some!

 

[chrisr999]

if you had the patience to look closely enough, Hurkyl, maybe you'd get it.
I write mathematics courses and train teachers. Honestly, you'd think a serious student would want to know exactly how it works.
What I explained is perfectly consistent.

 

[jgens]

I guess my average score of 100 per cent in higher level mathematics means nothing to some!

I could also argue that many mathematicians' insistence that derivatives are not fractions means nothing to some (note: I'm not a mathematician). To be honest, every calculus book that I've read (granted I haven't read a lot of them) has made a point that derivatives are not fractions (when dealing with real numbers only).

 

[chrisr999]

Don't get bogged down with whether you are dealing with real numbers, infinitesimals or fractions. Understand the "geometry" behind it. You already know that the real number system represents a continuum of scale. Differentiation starts out by locating 2 points on a curve and calculating an average rate of change, then working towards an "interval of zero" which you can discard and think in terms of the tangent of the angle the tangent is at.

 

[chrisr999]

I've seen that myself jgens, and students end up quite confused, those errors are also normally accompanied by numerous other inconsistencies, often using examples that are out of context.

many textbooks are not written by mathematicians.

Many maths teachers are not mathematicians at all, I've seen so many try to correct maths papers handed in by students and you may not believe what I've seen.

I am a mathematician.

You cannot understand calculus without examinig closely what's occurring at those limits

 

[jgens]

Don't get bogged down with whether you are dealing with real numbers, infinitesimals or fractions. Understand the "geometry" behind it.

I certainly understand the geometry behind derivatives and I understand intuitively the concept of a ratio between two infinitessimal numbers. However, moving beyond intuition, it's important to understand the mathematical princples behind calculus. In the real number system, infinitessimals do not exist. Unless you want to rigorously define infinitessimals, work out all of their properties and develop calculus from that number system (I think this is done in nonstandard calculus) you have to abandon the notion of a ratio of infinitessimals.

 

[chrisr999]

the "value" of an "infinitesimal" is an element of the set of positive quotients, which is a subset of the set of real numbers. The "infinitesimal" may be changing in value as you approach a tangent in the case of derivatives. It may even be disappearing, but that's because of how derivatives are "derived".
Check the alternative view of the geometry of the tangent.

Remember the number systems are "representations".
You can work with negative values also but remember you are working with little "lengths" on a diagram when learning differentiation.

You calculate the "value" of the derivative at a real value of the variable.
That locates where you are calculating the specific rate of change of the function at a specific value (real), but the derivative itself is being calculated by examining, in terms of the function equation, the tangent of the angle of the tangent, in other words, the "ratio" of the sides of the "vanishing triangle" which you can magnify and observe it's structure flush against the tangent in the diagram in the .pdf file attached previously.

 

[chrisr999]

Also, "infinitesimals" are just small measurements of the length of a line, that's all.
Don't be concerned about understanding their "properties".
They are lengths on the real number line from zero up to some small value if you want to look at it that way.
Hair widths, tiny differences between two positions, molecular dimensions, whatever way you like to look at it.
Examine as much as you can independently.

 

[jgens]

the "value" of an "infinitesimal" is an element of the set of positive quotients, which is a subset of the set of real numbers.

Really? Infinitessimals are not part of the real number system so they cannot be a subset of the real numbers. Here's how I think that we can show an infinitessimal is not a real number: Suppose k is a positive infinitessimal, then 0 < k < 1/n ∀n ∈ N. However, using limits we can show that there is no positive number k with this property hence, k = 0. While I didn't prove my statement with limits, if you've taken calculus you should be able to do that on your own.

Since this seems to be along the lines of what you're discussing, I figure I'll post these here. Notice that in order to do calculus with infinitessimals you need to introduce several new concepts.

http://en.wikipedia.org/wiki/Hyperreal_number

http://en.wikipedia.org/wiki/Non-standard_calculus

 

[jambaugh]

Here is how I explain this issue with my calculus students. At first we define the derivative which is not a fraction but a limit of a fraction. We prove various limit laws of which the important ones are:
Limits of fraction are fractions of limits (if all limits and expressions are defined).
Limits of products likewise (and especially limits of products of fractions for our purposes)

We then learn the chain rule and observe that though it is not cancellation of fractions mathematically the Leibniz notation for the chain rule seems to follow the form of fraction cancellation. This is a virtue of the Leibniz notation, it does some of our thinking for us by allowing an old algebraic rule, fraction cancellation, to keep track of the new calculus rule, the chain rule.

All the while I am strongly hinting that when we talk about differentials we will better understand the Leibniz notation as being equal too though not defined as a fraction.

Finally we discuss differentials as new variables constructed from old variables which have specific relationships inherited from the relationships we impose on the parent variables. In particular if y = f(x) then dy = f'(x) dx and thus by the definition of the differentials the derivative is indeed equal in value to the ratio of the differentials dy/dx.

We then discuss the differential operator which acts on an expression which we may equate to an implicit dummy variable. The differential of the expression yields the expression which must be equal to the differential of that implicit variable. For example
d(x^2) is just du for u=x^2 and thus du = 2x dx.

I think that we can resolve the "derivatives are fractions" "no they are not" arguments by distinguishing the definition of the derivative (which is a limit of a fraction) from the definition of a differential (which defines them as variables whose ratios are in fact derivatives).

I finish my discussion on differentials by having the students consider the equation:
\frac{dy}{dx} = dy \div dx
The left hand side is Leibniz's notation for a derivative and the right hand side is the ratio of two differentials. Totally different types of objects on each side and the equation itself is not a tautological identity but rather is an implication of the definition of the differentials.

For my students we then revisit the chain rule with this in mind. I find they get a much clearer understanding both of the power of Leibniz notation and in the distinction between differentials and derivatives from this picture. They are thus much better prepared to dive into the integral notation which we hit in the last third of the semester of Calc 1. Not to mention their better ability to deal with implicit differentiation and related rates.

EDIT: Notice that by defining differentials as variables (relative coordinates for a point on a tangent line) there is no need to invoke infinitesimals.

 

[jgens]

Don't be concerned about understanding their "properties".

Here's the problem with this. If you don't fully develop infinitessimals, how can you rigorously develop calculus and how can you be certain that their are no obvious inconsistencies?

 

[jambaugh]

Really? Infinitessimals are not part of the real number system so they cannot be a subset of the real numbers.

You can work with infinitesimals as real numbers. However you need to use an alternative equivalence relationship in the "equations" involving differentials as "infinitesimal quantities".
Allow differentials to be variables which linearly depend on an implicit parameter (we can call epsilon) and treat equations involving differentials as equivalence relations relating limits:

f(x,y,dx,dy) \doteq g(x,y,dx,dy)
translates to:
\lim_{\epsilon \to 0} \frac{f(x,y,dx,dy)}{g(x,y,dx,dy)} = 1
where epsilon is that implicit parameter to which all differentials are assumed to be proportional.

With this treatment there is no need to invoke exotic esoteric "hyper-super-duper-real numbers".

In short the problem with differentials as infinitesimals is not in the infinitesimal concept itself but rather in the sloppy use of = in equations.

 

[jgens]

jambaugh, is it possible to rigorously develop calculus using differentials? I thought that they were more a calculational trick than anything else, but perhaps not?

 

[chrisr999]

Some mistyping is corrected for the .pdf file.

You can consider that they are already developed as they are simply small measurements, with both heading to zero at the same time (the numerator and denominator of the ratio).
The tangent is always there, but you see, the reason we've ended up with the "infinitesimals" is because the "gradient of a line" is vertical distance divided by horizontal distance for a right-angled triangle placed against the line itself, no matter what size the triangle it is since the ratio of the vertical side to horizontal side is constant (tan(angle)), and this "crude" line gradient is then being "tweaked" to rest on top of the tangent.
Differentiation starts out with an "approximation" to the tangent gradient by choosing two points on the curve. This is because we only have the function equation to deal with!
but tilting this line until it "becomes the tangent" cause the numerator and denominator of the line gradient to start disappearing. Examine the diagram in the .pdf file... this is what is happening with the triangles to the right of the tangent. They are there to focus you on how we end up having to evaluate a limit FROM THE FUNCTION EQUATION.
Once you see the triangle disappearing, you switch to the triangles on the left of the tangent, to see exactly what the ratio of the sides are at the limit! It doesn't matter which triangle you examine! the ratio of the sides of the triangle you cannot see is the ratio of the sides of any of those triangles with the angle theta.
You won't be able to understand it without realising why we started out with the triangles on the right of the tangent and ended up with the triangles on the left.
Once you comprehend it visually, you don't need to think in terms of number systems.
Remember "any" number can be "expressed" as a fraction anyway.
Try to "visually understand this using the mathematical way to write a line gradient.
Remember 2 times multiplications can be represented visually as a line through the origin with a gradient of 2 and so on.

 

[HallsofIvy]

Here's the problem with this. If you don't fully develop infinitessimals, how can you rigorously develop calculus and how can you be certain that their are no obvious inconsistencies?

Calculus textbooks from the middle nineteenth century to about 1960, and almost all since 1960, do NOT use infinitesmals to "rigorously develop calculus"- they use limits. Starting about 1960, a new treatment of infinitesmal based calculus was developed as "non-standard Calculus". You won't see it in introductory calculus texts because it uses very deep results from Logic and "model theory" to define "infinitesmals".

jambaugh, is it possible to rigorously develop calculus using differentials? I thought that they were more a calculational trick than anything else, but perhaps not?

Differentials are rigorously defined and used in diffential geometry.

 

[jgens]

Geometically the derivative is not a difficult concept and visually I understand what it means. However, a geometric picture is not good enough in mathematics and when formalizing concepts with rigorous proof, you almost need to forgo geometric intuition completely. This is when number systems come in handy.

 

[jgens]

Calculus textbooks from the middle nineteenth century to about 1960, and almost all since 1960, do NOT use infinitesmals to "rigorously develop calculus"- they use limits. Starting about 1960, a new treatment of infinitesmal based calculus was developed as "non-standard Calculus". You won't see it in introductory calculus texts because it uses very deep results from Logic and "model theory" to define "infinitesmals".

I actually posted a link to the non-standard calculus wiki! My point to chrisr999 was that he had not even developed nor mentioned a rigorous definition of infinitessimals so he could not be sure they were consistent.

Differentials are rigorously defined and used in diffential geometry.

Cool! I learned something new today!

 

[chrisr999]

What we mean by the "limit of a fraction" jgens, is depicted visually on the diagram in the .pdf file. It's a bit like putting a folding ladder against a window. You hoist it and unfold it until it rests safely against the window.

The fraction in question is first the gradient of a line joining two points of a curve.
this gradient does not give you the derivative at the point of tangency! but it's a start as we can use the gradient of a line equation to get going.
then, you bring the second point over to the first. "Unfortunately"! or fortunately in the case of the curious student, the triangle we were working with vamoosed.

At the limit the gradient is the tangent gradient in spite of the fact that the triangle imploded.
If you understand it geometrically, you'll see it all soon enough.

then you will find the mathematical language no different to English.

 

[chrisr999]

No, jgens, the more clear the geometry, the more you realize what is being written in the mathematics language.
Derivatives can be fully understood geometrically, at that point you can "invent" your own mathematical symbols to describe it, but as things stand you should try your best to understand why Leibniz wrote the notation using the format of a ratio.
He did it for a clear reason. There was no notation until it was developed to "describe" whatever it was that was being communicated.

That's how mathematics notation is developed. The geometry, or whatever is being analysed, is primary to the language

 

[chrisr999]

you see, jgens,
calculus examines CURVES using STRAIGHT LINES.

The "infinitesimals", which is a term i haven't used until this discussion!, go hand in hand with the "limits".

You can't have rain without water, they are the same thing.
We are comfortable with straight lines.

Geometrically, calculus uses straight lines to handle curves and in order to be exact, not just "roughly accurate", the length of those straight lines go to zero!
The trickyness of the math handles this vanishing act!

sounds really contradictory, but it's pretty clever. these mathematicians were good.
It sounds more complicated than it is, it just takes getting used to, but without your mental microscope, it may not seem so clear

 

[jgens]

Again, I understand the derivative geometrically! What I'm saying is that a geometric picture means nothing in terms of proof nor does it mean anything in terms of the derivative as a fraction. If you want to work with differentials as jambaugh suggested or non-standard calculus as both HallsofIvy and I mentioned then there is nothing wrong with saying the derivative is a ratio of dy to dx.

If however, you're working within the real number system, the notion of the derivative as a ratio of two infinitessimals is non-sensical because infinitessimals do not exist as a subset of the real numbers.

Proof: For definiteness, suppose k is a positive infinitessimal, then k must satisy,

0 < k < 1/n ∀n ∈ N

This clearly defines a sequence. Letting the sequence {x_n} = 1/n we can show that this sequence is convergent to zero using the definition of convergence.

(∀ε)(ε > 0)(∃N)(∀n)(if n > N then 0 < |x_n - 0| < ε)

If ε > 0, then clearly for some N ∈ N, 0 < 1/N < ε. However, if n > N then x_n = 1/n < 1/N < ε, hence, 0 < |x_n - 0| < ε.

Consequently, 0 < k < 0 and by the Squeeze Theorem, k = 0. Q.E.D.

We've argued this point ad nauseum!

 

[chrisr999]

You're getting too deep into definitions and "ways of description", jgens.
Sincerity to understand will unravel it for you very simply.
Keep trying, no-one's arguing about anything!
Look simply and you will see that the maths in the situation you are trying to understand has a geometric or "point co-ordinate" foundation to it!
Sometimes previous knowledge can get in the way of seeing something and no-one's co-ercing you to adopt their point of view!
Only, "have a look at this, do you see what i see?"

 

[chrisr999]

If you refer back to my .pdf diagram,
are you seeing the ratio of the "infinitesimals" or the ratio of "dy" to "dx" at the point "t" as being EXACTLY equal to the ratio of the 3 triangles on the left side of the tangent to the graph at "t"?
Check if this is clear first.
Remember, you cannot see the infinitesimals at the limit, but the overlapping triangles show you what's happening to the "infinitesimals".
I'm referring to the "infinitesimals" as the perpendicular sides of the triangle that is imploding as the "crude gradient" line is being tilted to reach the tangent.
Because the triangles to the right are disappearing.
Therefore the tangent reveals the ratio of the infinitesimals.
Using the tangent, you don't even have to bother with infinitesimals!
But if you can "see" this, you'll see the whole picture of what happens at the limits and not just a part of it.
Notice that textbooks tend to concentrate on one description or another.
I've encapsulated the entire story of a derivative, meaning the "instantaneous rate of change" of the graph with this picture.
I designed it today, it's not from any textbook, then i made a .pdf file.

Persist and if you go beyond the frustration of trying to understand, it will reveal it's intricacy to you, maybe subconsciously at first but it will become clear unless you cut off your enquiry.

 

[chrisr999]

Try to imagine the triangle that's touching the graph at "c" and "t" getting smaller and smaller as "c" moves to "t".
It's disappearing but the limit of the ratio of it's perpendicular sides, as "c" continues on to "t" is the ratio of the perpendicular sides of any of the triangles at the left of the tangent.
This is what's happening.
To understand it all, you only need appreciate the following...

Why start at "a"?
Why move "a" to "t"?
Why start the maths using the gradient of a straight line?
Why approach a limit?
Why is the ratio of the sides of the disappearing triangle equal to the ratio of the 3 lined up on the tangent?

Imagine the point "a" moving to "t" and imagine what the triangle "looks like" as "a" hits "t".

There is no triangle right! at least there shouldn't be. not quite!
mathematically, if you are driving your car, you are actually traveling at an instantaneous speed at an instant in time, of course that instant has no time interval!

You see in school, students are taught to work with averages, then move on to calculus and at best many can perform the techniques without grasping the intricacy of calculating with an interval of zero,
you see the same in integration.
Don't get trapped in the definitions of limits, derivatives, differentials, infinitesimals, real numbers etc, otherwise you're getting into "humpty dumpty" territory. Try to get the "story" of it. That's what mathematicians have to do.
If someone hires you to get an "A" for their "D" level student, you don't use the same teaching materials that got them a "D".

How long is the moment we live in?
it can't be measured.
You can only measure an "interval"

What's the probability of being a height or age if that height or age is a real value?
you can't measure it, hence we use probability distributions and analyse intervals instead.