Linked Questions

67
votes
10answers
12k views

Why can't the second fundamental theorem of calculus be proved in just two lines?

The second fundamental theorem of calculus states that if $f$ is continuous on $[a,b]$ and if $F$ is an antiderivative of $f$ on the same interval, then: $$\int_a^b f(x) dx= F(b)-F(a).$$ The proof of ...
14
votes
6answers
2k views

Differentials Definition

Please define differentials rigorously such that they give a consistency to their use in the following links. I have read Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? What is the practical ...
5
votes
3answers
2k views

When can't $dy/dx$ be used as a ratio/fraction?

By searching this question, I found this: Can I ever go wrong if I keep thinking of derivatives as ratios? However, the answers don't have what I'm looking for! (Edit: Meaning, a counterexample. ...
8
votes
1answer
1k views

Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
5
votes
3answers
338 views

Learning differential calculus through infinitesimals

In class, we've studied differential calculus and integral calculus through limits. We reconstructed the concepts from scratch beginning by the definition of limits, licit operations, derivatives and ...
18
votes
1answer
889 views

Why do we treat differentials as infinitesimals, even when it's not rigorous

From single-variable calculus where we first encounter differentials we are told fairly often that differentials are not to be treated as infinitesimal quantities/objects (but we are never really told ...
3
votes
1answer
372 views

Differential and Infinitesimals

In a calculus textbook I have (Calculus, Stewart), it states that for a differentiable function $y=f(x)$, the differential of the function is defined as $$dy=f'(x) dx.$$ It states that $\Delta x\...
2
votes
2answers
190 views

What justifies writing the chain rule as $\frac{d}{dx}=\frac{dy}{dx}\frac{d}{dy}$ when there is no function for it to operate on?

This previous question of mine has lead me to ask the following question: It was my understanding that the chain rule $$\dfrac{du}{dx}=\dfrac{dy}{dx}\dfrac{du}{dy}$$ only makes sense when there is ...
2
votes
3answers
143 views

What is the simplest, yet still rigorous, way to define $\text{d}x$?

While reading calculus books, I see sections on differentials which refer to "infinitessimals" in a very loose way, alluding to the fact that this view on calculus is not the standard, but makes a lot ...
3
votes
1answer
248 views

What are the grounds for treating 'dx(differential, infinitesimal)' as if they were numbers?

I'm studying calculus and sometimes I find it strange to treat dx(differential) like numbers! Substitution rule would be a good example. ( I will use the first example in this website http://tutorial....
0
votes
2answers
100 views

Is $~f(x+dx)=f(x)+df ~$ true? How to prove it?

It's true that $f(x+\Delta x)\approx f(x)+\Delta f$, but is it still correct that $f(x+\mathrm{d}x)=f(x)+\mathrm{d}f$ ? If so, what's the proof? Since $\mathrm{d}x$ represents an infinitesimal ...