Hardest part of calc 3?

Thought calc 2 was easy. What is the hardest part of calc 3?

Other urls found in this thread:

cut-the-knot.org/wiki-math/index.php?n=Calculus.DerivativesWithoutLimits
math.hawaii.edu/~heiner/calculus.pdf
terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/
en.wikipedia.org/wiki/Transfer_principle#Transfer_principle_for_the_hyperreals
twitter.com/SFWRedditVideos

[math] ∫_ { ∂Ω } ω = ∫_Ω dω [/math]

> Calculus: The Equation

I swear people who say calc 2 was easy lucked out and had an easy professor. One could say "Well, doesn't that apply to every class?" No. I would much rather have a 'hard' calc 1 class than a hard calc 2 class.

I currently am an SI leader for calculus 2. The amount of people this class fails and retakes I've seen is ridiculous. There is too much stuff thrown at you at once compared to calc 1 and 3.

for non americucks, what does calc2 teach you?

Calc I: Differentiation, intro to Integration
Calc II: Integration
Calc III: 3D and Multivariable Calculus

thanks

Also limits and series in calc 2

Sequences and Series also go in calc 2 (at least where I'm from)

We covered limits at the end of Precal, and like the first few days of Calc I.

what? It's not some intelligence thing. The general derivation of the derivative is the limit of the difference quotient. Sure, you can just jump straight to rules and never learn those, but that's not going about it a smart way, it's skipping the fundamental theoretical background for how you can actually find a derivative.

>How did you do derivatives without limits?
By memorizing patterns of symbolic manipulation without any real understanding of the underlying concepts.

My class pretty much went here's the whole deal on how to calculate derivatives using limits the hard way. Do that for a couple homework assignments and then dropped it.

>google: derivative without limits
>cut-the-knot.org/wiki-math/index.php?n=Calculus.DerivativesWithoutLimits
>math.hawaii.edu/~heiner/calculus.pdf
>We introduce di erentiability as a local property without using
limits. The philosophy behind this idea is that limits are the a big stum-
bling block for most students who see calculus for the rst time, and they
take up a substantial part of the rst semester. Though mathematically
rigorous, our approach to the derivative makes no use of limits, allowing
the students to get quickly and without unresolved problems to this con-
cept. It is true that our de nition is more restrictive than the ordinary one,
and fewer functions are di erentiable in this manuscript than in a standard
text. But the functions which we do not recognize as being di erentiable
are not particularly important for students who will take only one semester
of calculus
how's sophomore year

This is the stupidest thing I've ever seen. It's literally for kids to retarded to understand a limit. The other posters were right to say that you need limits to define a derivative, because you do.

This is how my class operated too. It could be a bit tedious, but it's important to understand the conceptual means of finding the derivative and not just memorizing symbolic manipulations.

I'm in my last quarter at a community college, so we've got 4 calculus classes because of the quarter system. I'm assuming anything in calc 4 here would be in calc 3 somewhere else. Anyway, for me the hardest part of calc 4 is the triple integrals over shapes stuff. I think it might just be my professor, he's some pure math faggot who skips virtually all applications. He once described density as "some sort of mass distributed over some quantity of space". I swear to god he just jacks off to those gay ass shapes in his office and locks the door. He sounds exactly like the teacher from Beavis and butthead and it makes me want to sleep. His last test 6/9 questions were triple integral shape questions.

>This is the stupidest thing I've ever seen. It's literally for kids to retarded to understand a limit.
>moving the goalpost this hard
they implied derivatives can't be defined without limits, clearly they can.
>The other posters were right to say that you need limits to define a derivative, because you do.
i just linked to several examples of it. just because you don't like it doesn't mean it's not true you fucking SJW.

yep

probably green's theorem and that was only because it was taught to me in a very rushed manner without proof.

Not who you're replying to, but do you understand the thing you linked to? Being able to define the derivative for a limited subset of differentiable functions is not the same as the full definition of the derivative. They are correct in saying that the general definition of the derivative cannot be given without limits. And don't go off on some autistic rant about how them not directly saying the word general means they could have just been referring to that. When someone says "definition of the derivative" with no qualifiers, they don't mean "definition of the derivative for some but not all of functions for which a derivative can be defined".

what the hell i learned the calc II stuff in regular calculus I in high school

This

How the hell can any professor worth his salt teach people Calculus when they don't even understand the basic concept of a limit -- which leads of course to the discovery of the concept of the derivative?! What a joke.

You're the one being autistic and arrogant. You're assuming because you were taught about a concept a certain way, that's the way is has to be defined. Have you people even taken a math class past the 300 level? There are different ways to formulate the same concept and as long as they are mathematically rigorous they are perfectly valid. Take for example the constuction of the number line, you can the methods of non-standard analysis which includes hypperreals, standard analysis which uses the real number system, that wildberger shit that gets spammed here, or any other construction so long as it's mathematically sound. Educate yourself before speaking on a topic.

High school calculus 1 is the same length or longer than 2 semesters of college calc.

Usually they give rigorous definition and properties of the Riemann integral. You went to a pretty good high school, user.

But the limit definition isn't just the traditional way to define the derivative, it is the ONLY way to consistently define the derivative for all differentiable functions. I completely understand and agree everything you say about multiple definitions being acceptable if they are rigorous, but the definitions without a limit are either a) not rigorous or b) only work for a subset of differentiable functions. Can you give a definition of the derivative that is rigorous and does not use limits but defines the derivative of any differentiable function f(x)?

Infinitesimals, I guess. Not really popular though, the full details are kind of intrincate.

Uh, the reason that all the constructions of the real numbers are accepted is that they all yield the same result. You can use hyperreals, Cauchy sequences, Dedekind cuts, etc. and you always get the same real numbers. This is not true for your method of defining the derivative, as it does not give a derivative for some differentiable functions.

I don't think that you can actually define the derivative with infinitesimal, they're just connected.

Pretty sure you can. See exercise 4 here:
terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/

I did a bit of research on this and everything I could find just defines the derivative with the same form as the limit definition but instead of explicitly invoking a limit, uses the definition of a limit through infinitesimal. You might consider this to be distinct, but I don't think I would.

>it is the ONLY way to consistently define the derivative for all differentiable functions
it's not the only way. and it's a bad argument. You're assuming that the definition of derivatives using limits IS the definition. just because a different formulation applies to a smaller set doesn't mean it's not valid, as i'll explain.
>Can you give a definition of the derivative that is rigorous and does not use limits but defines the derivative of any differentiable function f(x
[math]f'(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x}[/math]
where st() is the standard part function and delta-x is an infinitesimal. aka, the definition of derivatives in non-standard analysis.
now this definition applies to more functions than the definition in standard analysis, so would you say the standard analysis formulation is wrong? Of course not.

see

But that use of the infinitesimal change and st is just the infinitesimal definition of a limit. You've just removed the limit and replaced it with a definition of the limit, it's the same thing.

Also, I never said that defining differentiability on a smaller set of functions is necessarily invalid, just that it doesn't constitute a full and general definition of the concept of differentiability.

Not the guy you are arguing with, but does this work for partial derivatives and more generally multivariate calculus? It doesn't seem like it would but I haven't spent much time on nonstandard analysis so I'm not sure.

this shit doesn't even work in [math] \mathbb{R} [\math] genius. you have to extend your shit to something else (the hyperreals) which isn't what you usually use.

when someone starts talking about nonstandard analysis to justify shaky notions you know they're full of shit

>But that use of the infinitesimal change
it's not infinitesimal change. it's an actual infinitesimal. that's the whole point of non-standard analysis.
>st is just the infinitesimal definition of a limit
it's not. it's a rounding function which has a relation which limits but is not defined by it. In fact, you can define the limit in non-standard analysis using the standard part function. A stronger definition is formed by using dedekind cuts.
>just that it doesn't constitute a full and general definition of the concept of differentiability.
but again, you're assuming the limit definition of derivatives when you say that. Think about what you mean by full and general. Again, if there was a function in non-standard analysis that was not differentiable by using a standard analysis version of a limit, would you say the standard analysis version is wrong?
>The theorem to the effect that each proposition valid over R, is also valid over *R, is called the transfer principle.
en.wikipedia.org/wiki/Transfer_principle#Transfer_principle_for_the_hyperreals
yes. and i would imagine any theorem in multivariate calc could be generalized to the hyperreals.

>this shit doesn't even work in [math] \mathbb{R} [\math] genius
that's the point you idiot
>you have to extend your shit to something else (the hyperreals) which isn't what you usually use.
I clearly stated that's what I was doing
>aka, the definition of derivatives in non-standard analysis.

>when someone starts talking about nonstandard analysis to justify shaky notions you know they're full of shit
try reading the thread before responding. I was making the point that one formulation does not invalidate another just because it's stronger in the same way that non-standard analysis does not invalidate standard analysis.

someone said they didn't learn any rigor in calculus, for example, no limits
someone else said that's dumb, you can't do analysis of a metric space without understanding the topology of a metric space
then you pop out of nowhere and bring up something grossly unrelated which seems to resemble calculus for basic applications but nothing more

how the fuck can you think you're making a point here? what the fuck is your point?

>en.wikipedia.org/wiki/Transfer_principle#Transfer_principle_for_the_hyperreals
Nah, that applies to stuff expressible in first order in the language of ordered fields, right? Most theorems in real analysis aren't like that.

For me, calc 1 was
>inverse trig integration/differentiation
>l'hospital's rule
>volume of solids by revolution
>trig substitution
>integration by parts
>series and sequences convergence tests
>basic conics and ellipticals
>intro to parametric equations

Okay, a couple things. One, you totally took the st quote out of context. I'm aware that st() gives the standard part of a hyperreal number. I didn't say it was a limit, I said in the specific context, it was part of a construction that was functionally identical to a limit. Also, non-standard analysis cannot be used for multivariable calculus.

yeah, this person is totally incomprehensible. Like, it sucks that your calc classes did a shitty job of teaching you, but that doesn't mean that calculus is somehow wrong or real analysis shouldn't use limits.

>Also, non-standard analysis cannot be used for multivariable calculus.
Why do you say so?

You can look it up and see that this is true. Really, differential forms are the best way to do this with multivariable.

>make claim
>get asked why
>"oh you can look it up"

provide evidence or don't say anything

>You can look it up and see that this is true.
Come on. This and nothing is the same, what source or arguments are you using?
>differential forms are the best way to do this with multivariable
I see no reason why a nonstandard version (or something that does the same) of this shouldn't exist. In fact the rigorous definition of differential forms seems already quite contrived, maybe there is some more natural nonstandard formulation.

if the definition of differential forms seems contrived to you and a fucking extension of ZFC with whacky shit all over the place seems "natural", you probably don't know mathematics

Why can't you? From my limited understanding any theorem in zfc is valid in ist. Doesn't that mean anything that can be formulated in zfc can be forrested in ist?

triple integrals are super easy fampai

pauls online notes explain how to do it.

Oh shit, I forgot that defining differential forms might require more math knowledge than a freshman would have. Sorry, I'll stick to something easier next time.

Wah? Nonstandard analysis is usually done in ZFC senpai. You just need and ultrafilter.

No way is stokes the hardest part of any class. Even in a class using spivaks calc on manifolds, it's just one of the last things in the book, not the hardest.

Literally neither of those links defined derivatives...

Differential forms are one of the most natural constructions in maths...

sorry calc 2

Just staring calc 3 next tuesday

I can see how one would arrive at that conclusion, as my own lack of knowledge of non-standard analysis tells me that more complexity is needed for multivariable. However, I would also assume that someone who actually is interested in non-standard analysis is also smart enough to realize this and attempt to add additional complexity.

So, seeing as you are implying you know more than the average person about non-standard analysis, please explain why it fails instead of suggesting "look it up" with zero relevant terminology to actually search for.

Alternatively, if your argument is "the wiki page for non-standard analysis does not contain the word multivariable", please don't promote something as fact when your ignorance is equal to ours.

My university has single variable calculus and multivariable calculus. I live in Canada
Single variable covered the following:
>review of functions & trig
>limits
>derivatives
>derivatives of exponents and logs
>curve sketching
>indefinite and definite integral, riemann integral
>integration by substitution, parts, partial fractions, product of sines and cosines, trigonometric substition,
>improper integrals
>area between curves
>volume of a solid of revolution

i'm doing what i assume is the equivalent of calc 3 (studying in australia, this is the 3rd calculus course ive taken) and the hardest things so far have been:
- fourier series
- laplace transforms
- power series solutions to DE's
- divergence and curl
- stokes theorem

Sounds more like diffeq

solids of revolution probably.

the issue is that it is a highly pragmatic and useful tool

jumping between series and standard integrals with difficult substitutions (4+ compounded trig substitutions) is what causes the issue, I think.

How is calculus even that
hard?

I mean I am just breezing through geometry and algebra like nothing and haven't been stumped yet.

Proving anything is easy once you memorize all the theorems, postulates, terms and definitions