## How hard is college level mathematics?

For the discussion of math. Duh.

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Kurushimi
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### How hard is college level mathematics?

So, throughout all of high school, I've been pretty good with math, earning straight A's for every math class I've taken. It's all been pretty easy. I imagine many people here have had the same experience. My high school also offers Multivariable Calculus and Linear Algebra, which I'm taking this year, my senior year, and they also seem to be the typical level of easy that math has been so far. What I was wondering is, for anyone here, when did math start getting harder in college? What about it was harder? Are there any good habits I should be forming in order to prepare for harder mathematics down the road?

mike-l
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### Re: How hard is college level mathematics?

At some point you stop calcuating things and start proving things. That's generally when it gets hard. You'll see the odd proof in calculus and linear algebra, but it's still pretty calculation based. Depending on your school, it might be a discrete math course, a group theory course or a real analysis course that really introduces you to proof based mathematics.
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gorcee
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### Re: How hard is college level mathematics?

It really all depends on the college, the curriculum, and your focus.

Most top-tier schools have "weeder" courses. Of course, they're never advertised as such, but these courses are designed, and the course syllabuses are planned, in such a way that students who slack off will not pass. Most technical majors have a few courses like this. In computer science, for example, Data Structures is typically a second year course that kicks the shit out of any student who doesn't take school seriously. So, you can expect at some point in your college career to encounter a class that will be more difficult than what you've experienced before.

However, the difficulty in these classes is not inherent in the subject matter, but rather in the amount of time and discipline required to ensure that you get a good grade. So if you work hard, you'll learn the stuff, and you'll do fine (while this holds true always, some classes are designed to exploit this fact of life). Often times bright students in high school have a hard time in "weeder" courses because they never had to buckle down and get things done. I was very much this way... I never worked in high school, and when I got to college, I was in for a rude awakening.

A second thing to consider is that in many colleges, the teachers are professors, and those professors are career professionals in their fields (mathematics, physics, etc.). In high school, your teachers are career teachers. So while high school teachers may be brilliant, they don't have to submit to the rigor of peer review, of grant writing, of tenure, PhD committees, working groups, industry/government consulting, etc. They have to be able to connect with a student and teach him the material to prepare him for the next step. So the expectations of what you will be taught and what you will learn are somewhat different in college. It's a higher level of rigor on the part of the student. In many of my aeronautical engineering classes, for instance, we never got partial credit, even if we did something silly like write 2+2 = 5 (also, the questions were cumulative multi-part, and if you got part a wrong, part b was guaranteed to be wrong, and so on). Was this hard? Yes. Was it hard because of the material? No. It was difficult because we were expected to know all of the little things, remember how to integrate various functions, etc.

As you get deeper into mathematics, there is a greater reliance on looking for subtle features/properties of the math to be able to move on. Can we do analytic continuation? Is this singularity removable? Can we ignore higher order terms? What does the Taylor expansion look like? Can we invoke AM-GM? What are the properties of the eigenvalues? It's not that it's naturally harder (because typically, when you see the solution, it's obvious), but rather that it requires more precision and more care.

Some math students get caught up in a multivariable class, or a differential equations class. Others manage to real analysis. It all depends. But I've advised dozens of students in my day, and one thing I tell them is that it's OK if you get to a point that's hard. Everyone hits a point like that sometimes. And once you get through it, it will get easier until you hit that point again. Learning is a road with many speed bumps, and they are not always in the same place for everyone. The only real advice I can give you is to take it seriously and be patient.

z4lis
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### Re: How hard is college level mathematics?

As you might gather, up to this point you've not actually done a whole lot of real mathematics. (Most likely. There are always exceptions.) That is, you've probably not actually proven anything or learned about much mathematics beyond some basic real analysis, algebra on the reals, and some plane geometry. (By no means is this a bad thing. Well, it is, but it has to do with the school system and not you. ) Coming out of high school, I was toying with the idea of going into physics and not mathematics simply because I didn't fully comprehend what mathematics was. I aced calc, and couldn't fathom there was much more to it. Graduate classes on algebra? Come on! What were those mathematicians really doing?

And then I hit the university, where I realized how much depth there was to mathematics. Everything I'd been exposed to, mathematically speaking, could be written up in about... twenty pages. The majority of what I'd been doing in my mathematics classes was just applications of a handful of theorems based on intuitive grounds and memorization of algorithms. That's what's so easy. The hard part of mathematics arrives when you don't have any examples to follow to solve your problems. You're tasked to prove a statement, and accomplishing that requires a combination of: sheer luck, time, being smart, and having a deep understanding of the material. Your ability to be creative and truly understand what you're doing become very relevant. The creativity kicks in because you must develop ways of understanding and proving statements, and the depth of understanding is required because, as someone else put very well, higher level mathematics requires a great deal of care and precision. If you aren't on top of what you're doing, it's very easy to overlook something in a proof.

Another point: there are two sorts of difficulty in a class. On one hand, there's a sense of technical difficulty in a class. (The sort that I don't like.) Technical difficulty concerns how hard the homework is graded, how much homework there is, how difficult the tests are, etc. etc. Then, there's the inherent difficulty of a class due to the subject matter. I've had classes that were incredibly easy, technically speaking, (there was little homework and the tests were not difficult) but the subject matter was horrifically complicated and dense. And then you have the opposite, things like "weeder courses" whose subject material is easy to understand, but the course is constructed to make you work hard to do well.

So, if you're asking about how hard a college level math course is, consider both of those. You could probably make a multivariate calculus course terribly difficult through grading schemes and homework assignments (long, messy iterated integrals come to mind). If you're wondering about when the inherent difficult of mathematics classes begins to ramp up, I'd say that, for all practical purposes, it will be during or shortly after you take your first course with a title along the lines of "Intro to Abstract Math" or "Intro to Proof-Writing", which I recommend you push yourself into as soon as possible if you're interested in learning mathematics! Even if you don't plan on actually getting a degree in mathematics, if you want to do serious reading in your free time or something like that, a class like that would let you begin to tackle the literature.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

Dark Avorian
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### Re: How hard is college level mathematics?

Hi, I'm sort of wondering the same thing the OP is, especially since I'm in something of the same boat. I've taken Calc and this year (junior) I'm taking "Abstract Algebra" which is taken from Gallian's Contemporary Abstract Algebra and focuses on proofs. We're on groups now and will cover rings and fields by the end of the year. So how does that kind of class relate to college level math classes if your talking about everything being related to proofs?
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achan1058
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### Re: How hard is college level mathematics?

Dark Avorian wrote:Hi, I'm sort of wondering the same thing the OP is, especially since I'm in something of the same boat. I've taken Calc and this year (junior) I'm taking "Abstract Algebra" which is taken from Gallian's Contemporary Abstract Algebra and focuses on proofs. We're on groups now and will cover rings and fields by the end of the year. So how does that kind of class relate to college level math classes if your talking about everything being related to proofs?
I do not know what that book has, but generally that class should be your typical "proof class".

Yakk
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### Re: How hard is college level mathematics?

Glancing at that book, one could teach an acceptable intro to rings/groups from that book.

The level of rigor matters hugely. If you covered every single chapter (even excluding the appendixes), and able to answer every problem given, and regenerate proofs to every theorem in those chapters, I'd be quite impressed with your high school. (that is an understatement)

At the same time, you can breeze over the hard parts and just give you a light tour.

The concepts of "what is a ring" and "what is a group" and "what is a finite" are relatively trivial. It is what you do with the definitions that gets interesting.

Spoiler:
BTW, what is "junior" year? http://en.wikipedia.org/wiki/Eleventh_grade ? (the year before the year you graduate?) Sorry, I'm a dirty furriner, I don't always get US education slang.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

achan1058
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### Re: How hard is college level mathematics?

Yakk wrote:
Spoiler:
BTW, what is "junior" year? http://en.wikipedia.org/wiki/Eleventh_grade ? (the year before the year you graduate?) Sorry, I'm a dirty furriner, I don't always get US education slang.
Wikipedia says Freshman, Sophomore, Junior, Senior as its order, and this can both be for high school or university students. I would think this is year 3 university.

Yakk
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### Re: How hard is college level mathematics?

He asks how his class relates to college level math. That would be a strange question for someone in college?

But ya, that is why I asked what it means...
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

the tree
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### Re: How hard is college level mathematics?

I still get confused when Americans say college and they mean what you do after being 18. I'd say math/everything gets harder as it gets more specialised. Rather than just learning 'math' you will have many completely seperate classes, there's not much relation between an introduction to real analysis and an introduction to discrete maths, even though the concept of proof is the same I found real analysis significantly easier than the discrete courses - I'm really weird by the way, for most people it's the other way around but the point is that there's probably already some parth of maths that you find harder than the other parts and there will be an entire course on just that thing, that is where maths gets hard.

gorcee
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### Re: How hard is college level mathematics?

In the US, college is post-secondary; that is, after high school. Most American students graduate high school at age 18. College is also (almost) synonymous with the European term "university." So an American student saying, "I'm in college," is the same as someone in Europe saying, "I'm in university." However, in the United States, "college" and "university" do have different meanings: a college is a 4-year post-secondary institution, so the highest accredited degree that it generally grants is a 4-year Bachelor's degree (BA or BS), and in specific programs, sometimes a Master's degree. A university, on the other hand, generally grants post-graduate degrees along with 4-year degrees. An "institute" can be a college or university that has a specialized focus, typically in engineering or the sciences (ie, MIT, RPI, RIT, etc.). I am not aware if this nomenclature extends overseas.

Our nomenclature for high-school is: Freshman (9th grade, typically ages 14-15), Sophomore (10th grade, 15-16), Junior (11th grade, 16-17), and Senior (12th grade, 17-18).

We recycle these terms for college. Freshman is 1st year, Sophomore is 2nd year, Junior is 3rd year, Senior is 4th year.

I would guess that Dark Avorian is a 3rd year high-school student (hence, junior, and referring to collegiate mathematics in the future, and no US college would teach calc as part of the normal curriculum to a 3rd year student).

voidPtr
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### Re: How hard is college level mathematics?

Dark Avorian wrote:Hi, I'm sort of wondering the same thing the OP is, especially since I'm in something of the same boat. I've taken Calc and this year (junior) I'm taking "Abstract Algebra" which is taken from Gallian's Contemporary Abstract Algebra and focuses on proofs. We're on groups now and will cover rings and fields by the end of the year. So how does that kind of class relate to college level math classes if your talking about everything being related to proofs?

You're really covering abstract algebra in a high school course? That's not a topic that's usually touched at my university until at least second year mathematics here (introduced in other courses like geometry or discrete structures) and not in depth until 3rd or 4th year. Sometimes I'm completely blown away by what topics people in middle and high school are learning. Not just because of the difficulty level but also because the topics are just so specialized and not likely to be relevant to anyone except people who are interested in theoretical mathematics.

gorcee
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### Re: How hard is college level mathematics?

voidPtr wrote:
Dark Avorian wrote:Hi, I'm sort of wondering the same thing the OP is, especially since I'm in something of the same boat. I've taken Calc and this year (junior) I'm taking "Abstract Algebra" which is taken from Gallian's Contemporary Abstract Algebra and focuses on proofs. We're on groups now and will cover rings and fields by the end of the year. So how does that kind of class relate to college level math classes if your talking about everything being related to proofs?

You're really covering abstract algebra in a high school course? That's not a topic that's usually touched at my university until at least second year mathematics here (introduced in other courses like geometry or discrete structures) and not in depth until 3rd or 4th year. Sometimes I'm completely blown away by what topics people in middle and high school are learning. Not just because of the difficulty level but also because the topics are just so specialized and not likely to be relevant to anyone except people who are interested in theoretical mathematics.

Often times, the high school coverage is somewhat watered down compared to a true upper-level college course. This isn't a bad thing, but I do think that the teachers/school oversell the material to the students a little bit. I often see people posting here and elsewhere that they are a high school student and they're studying, say, complex analysis and how they're doing really well and they love it all that, but really what they've covered is like, the first four days of a complex analysis course.

It's never a bad thing to expose students to mathematics, but I think it definitely causes a communications disconnect when those students ask questions/advice.

voidPtr
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### Re: How hard is college level mathematics?

gorcee wrote:
voidPtr wrote:
Dark Avorian wrote:Hi, I'm sort of wondering the same thing the OP is, especially since I'm in something of the same boat. I've taken Calc and this year (junior) I'm taking "Abstract Algebra" which is taken from Gallian's Contemporary Abstract Algebra and focuses on proofs. We're on groups now and will cover rings and fields by the end of the year. So how does that kind of class relate to college level math classes if your talking about everything being related to proofs?

You're really covering abstract algebra in a high school course? That's not a topic that's usually touched at my university until at least second year mathematics here (introduced in other courses like geometry or discrete structures) and not in depth until 3rd or 4th year. Sometimes I'm completely blown away by what topics people in middle and high school are learning. Not just because of the difficulty level but also because the topics are just so specialized and not likely to be relevant to anyone except people who are interested in theoretical mathematics.

Often times, the high school coverage is somewhat watered down compared to a true upper-level college course. This isn't a bad thing, but I do think that the teachers/school oversell the material to the students a little bit. I often see people posting here and elsewhere that they are a high school student and they're studying, say, complex analysis and how they're doing really well and they love it all that, but really what they've covered is like, the first four days of a complex analysis course.

It's never a bad thing to expose students to mathematics, but I think it definitely causes a communications disconnect when those students ask questions/advice.

Yeah, I suppose it may be a naming misnomer, like as you said, where an introduction to the complex plane gets interpreted as 'complex analysis'. But even then, something like 'multivariable calculus' taught at a high school level blows me away a little bit (not as much as abstract algebra mind you), unless it's a special school for gifted children or something. That's not normally taught until second year mathematics here..even 1st year honours mathematics is typically only single-variable with a higher-level of rigour added.I think introducing advanced subjects to high school students could really backfire if not taught properly.

Yakk
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### Re: How hard is college level mathematics?

There is nothing fundamentally hard about multivariable calculus.

Take functions that map R to R^2 and take its derivative. It is much like working on two single-variable problems. Maybe you go crazy and talk about integrating over arc lengths!

Now, talking about the 4th "derivative" of a R^4 to R^8 function as a tensor... but you don't have to do crazy stuff like that for it to be multi-variable calculus.

ObJoke: I mean, engineers do calculus. Clearly high school students can handle it!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

voidPtr
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### Re: How hard is college level mathematics?

I don' t know where everyone is from. But here, intro calc typically has very low grade averages. Whereas people who are mathematically inclined find it very easy, the general population does not and it's one of the reasons there are tons of jobs for tutoring it. In my grade 12th year we didn't even touch calculus until near the end and it was only a brief introduction to limits and derivatives.

mike-l
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### Re: How hard is college level mathematics?

People find it hard because there are lots of fractions, and they can't do fractions.

Only mildly joking.
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Yakk
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### Re: How hard is college level mathematics?

And you aren't allowed to cancel the d's!
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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++\$_
Mo' Money
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### Re: How hard is college level mathematics?

I've known people to get very confused when introduced to the general (multivariate) chain rule, either because they're not happy canceling [imath]\partial[/imath]s and [imath]d[/imath]s, or because they think${\partial u \over \partial x}{dx \over dt} + {\partial u \over \partial y}{dy \over dt} = 2{du \over dt}.$

Talith
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### Re: How hard is college level mathematics?

Yakk wrote:And you aren't allowed to cancel the d's!

dx/dx = 1 seems to be working for me... i can't seem to find any counter examples.

Spoiler:
I kinda stopped looking after I found that example >.>

achan1058
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### Re: How hard is college level mathematics?

mike-l wrote:People find it hard because there are lots of fractions, and they can't do fractions.

Only mildly joking.
I wanted to cry after reading this one. I TAed an engineer linear algebra course once, and the students are horrible when fraction shows up.

Yakk
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### Re: How hard is college level mathematics?

Talith wrote:
Yakk wrote:And you aren't allowed to cancel the d's!

dx/dx = 1 seems to be working for me... i can't seem to find any counter examples.

Spoiler:
I kinda stopped looking after I found that example >.>

dx^2/dx
Well, cancel the dx on the top and the bottom, and the only value left is 2. But x^2 is also x*x, so don't forget the x.

2x.

I guess it does work.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Kurushimi
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### Re: How hard is college level mathematics?

Yakk wrote:
Talith wrote:
Yakk wrote:And you aren't allowed to cancel the d's!

dx/dx = 1 seems to be working for me... i can't seem to find any counter examples.

Spoiler:
I kinda stopped looking after I found that example >.>

dx^2/dx
Well, cancel the dx on the top and the bottom, and the only value left is 2. But x^2 is also x*x, so don't forget the x.

2x.

I guess it does work.

Wait, if you cancel out the dx you either get x or if you want to be pushing it you get 2. You never get 2x.

GenericPseudonym
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### Re: How hard is college level mathematics?

Kurushimi wrote:
Yakk wrote:dx^2/dx
Well, cancel the dx on the top and the bottom, and the only value left is 2. But x^2 is also x*x, so don't forget the x.

2x.

I guess it does work.

Wait, if you cancel out the dx you either get x or if you want to be pushing it you get 2. You never get 2x.

This is actually a joke of sorts: dx^2/dx actually equals 2x in calculus. d/dx means "Find the derivative of", and the derivative of x^2 is, in fact, 2x.

Kurushimi
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### Re: How hard is college level mathematics?

I know, I'm saying the joke doesn't work because cancelling out the dx's doesn't leave 2x it leaves x.

WTFortissimo
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### Re: How hard is college level mathematics?

d(x^2)/dx = 2x
not 2, not x

do brackets that relieve some confusion?
you can't cancel it because d is not a coefficient as such, it's to be considered as part of an operation, as genericpseudonym says, d/dx means to differentiate something with respect to x.

Kurushimi
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### Re: How hard is college level mathematics?

no, I know how to take the derivative. That's not the point. He was comparing the way canceling out the dx's in dx/dx yields one just like taking the derivative of x, (d(x)/dx) yields one. This is not the same thing when you take the derivative of x^2 because the answer is 2x but canceling out the dx's yields x.

Talith
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### Re: How hard is college level mathematics?

Talk about beating a joke to death... I think we should forget it happened and get back to the OP topic.

arzaqi
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### Re: How hard is college level mathematics?

If you like challenges, then probably you should try some undergrad math contest. There are so many undergrad math contest, if you live in North America, you can try your luck at Putnam Math Contest. If you live somewhere else, you still can compete in IMC (International Math Competition) or ISOM (International Scientific Olympiad on Math). IMC and ISOM hold annualy. There you can measure your problem solving ability for undergraduate (or maybe graduate) math.

mdyrud
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### Re: How hard is college level mathematics?

If you're still in high school, there are also some math competitions you can take part in. The AMC is definitely the big one. None of these tests will really show you a great view of what college level maths is like, since they are testing your ability to see a neat solution to a problem, but they are an okay glimpse at some of the topics you will see. The Art of Problem Solving has a collection of problems and solutions to past tests. Definitely start with the AMC. The other tests listed below the AMC are tests that you qualify for based on your AMC score. If you have any questions on some of the solutions, go check out the forums. They discuss the solutions in depth there. Finally, depending on where you are located, check and see if your school has a Math League team.

wAriot
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### Re: How hard is college level mathematics?

I just registered in this forum to say the following: after reading the complete topic, I'm not sure if I'm feeling jealous or frightened right now. I mean, I'll be finishing high school this year, and I'll study Maths at the university. But here (I'm Spanish) we don't even have Calculus or Algebra at my level, just that, Maths (oh, well, does Statistics count as a subject?). So, it's really difficult to participate in any contest or to have a decent level of knowledge in this. When reading some posts, I just feel like an idiot. Ok, I know how to derivate, but WTF does ${\partial u \over \partial x}{dx \over dt} + {\partial u \over \partial y}{dy \over dt} = 2{du \over dt}.$ mean?. Sincerely, I respect, admire and envy all who understand that. (btw, as I said, I'm Spanish, so please, please correct me if i did any grammatical error, so I can correct it)

ImTestingSleeping
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### Re: How hard is college level mathematics?

wAriot wrote:I just registered in this forum to say the following: after reading the complete topic, I'm not sure if I'm feeling jealous or frightened right now. I mean, I'll be finishing high school this year, and I'll study Maths at the university. But here (I'm Spanish) we don't even have Calculus or Algebra at my level, just that, Maths (oh, well, does Statistics count as a subject?). So, it's really difficult to participate in any contest or to have a decent level of knowledge in this. When reading some posts, I just feel like an idiot. Ok, I know how to derivate, but WTF does ${\partial u \over \partial x}{dx \over dt} + {\partial u \over \partial y}{dy \over dt} = 2{du \over dt}.$ mean?. Sincerely, I respect, admire and envy all who understand that. (btw, as I said, I'm Spanish, so please, please correct me if i did any grammatical error, so I can correct it)

Here's some hard truth I learned quickly in my study of mathematics at university which I wish someone told me early on: What people claim to know, what they want you to think they know, and what they actually know RARELY line up nicely. I'm talking about your peers, your teaching assistants, and even your will-be professors.

The point I would like to make is that you will generally feel inadequate when you're studying math. There will always be someone who knows more than you. Please, do not let this discourage you. Try not to feel jealous or frightened (though those are normal feelings for someone about to enter university, I assure you). In my experience and from what I've discussed with others, you should feel uneasy and a bit overwhelmed with the amount of mathematics you don't yet know, and you should use that to motivate you to chip away at some more math.

(I just reread what I wrote and it reads a bit dramatic to me, but I do think it is important that you understand that what you're feeling is perfectly normal!)

Yakk
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### Re: How hard is college level mathematics?

wAriot wrote:WTF does ${\partial u \over \partial x}{dx \over dt} + {\partial u \over \partial y}{dy \over dt} = 2{du \over dt}.$ mean?

The ratio between the change in u to the change in x when you change x by small amounts, times the ratio in the change in x to the change in t when you change t by small amounts, plus the ratio of the change in u to the change in y when you change y by small amounts time the ratio of the change in y to the change in t when you change t by small amounts, equals twice the ratio of the change in u to the change in t when you change t by small amounts.

Roughly.

The [imath]{\partial u \over \partial x}[/imath] construct is basically that "ratio of the change in u to the change in x when the change of x is really small" construct. You can think of it as some kind of slope on some kind of graph (delta y divided by delta x): as it happens, for "nice" functions there is a natural "change when the denominator is small" value that has nice properties and lets you do interesting stuff with it.

And that is calculus. Well, there is much more to it, but that is the start of it.

Hope that helps.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

wAriot
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Location: Galicia, Spain

### Re: How hard is college level mathematics?

Yakk wrote:
wAriot wrote:WTF does ${\partial u \over \partial x}{dx \over dt} + {\partial u \over \partial y}{dy \over dt} = 2{du \over dt}.$ mean?

The ratio between the change in u to the change in x when you change x by small amounts, times the ratio in the change in x to the change in t when you change t by small amounts, plus the ratio of the change in u to the change in y when you change y by small amounts time the ratio of the change in y to the change in t when you change t by small amounts, equals twice the ratio of the change in u to the change in t when you change t by small amounts.

Roughly.

The [imath]{\partial u \over \partial x}[/imath] construct is basically that "ratio of the change in u to the change in x when the change of x is really small" construct. You can think of it as some kind of slope on some kind of graph (delta y divided by delta x): as it happens, for "nice" functions there is a natural "change when the denominator is small" value that has nice properties and lets you do interesting stuff with it.

And that is calculus. Well, there is much more to it, but that is the start of it.

Hope that helps.

I had to read it a couple of times, but I think I get it. Anyway, thanks for your replies. I still feel jealous (Multivariable Calculus? Linear Algebra? Not even in my best dreams)

mdyrud
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### Re: How hard is college level mathematics?

College math is different, but it isn't as intense as you may think. Especially Multivariable Calculus. You're essentially just taking what you know about calculus in one variable, adding a few more variables, and performing the same steps a few extra times.

Yakk
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### Re: How hard is college level mathematics?

Linear Algebra:
Let X be a collection of elements of some kind, where you can add the elements to each other in a way that acts like adding, and scale elements by the real numbers (or something sufficiently similar to the real numbers).

Let f be a function on X. If f( lamda a + b ) = lamda f(a) + f(b) always works, then f is called "linear". (lamda in this case is a scale factor -- like zooming -- and a and b are elements of X).

Linear Algebra is the consequences of the above. What else can you know about f (and collections of functions like f) given that you know that little bit about it?

Turns out the answer is "lots".
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

skullturf
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### Re: How hard is college level mathematics?

wAriot wrote:(btw, as I said, I'm Spanish, so please, please correct me if i did any grammatical error, so I can correct it)