I may have made the connection if we were using actual numbers and the multiplication would have been something like 1*10^(-5000), but they were arbitrary numbers, a and b. It was basically “show that for an arbitrary a and b, you can express it as <some equation>”. The value ab is tiny, and probably negligible, but I wasn’t convinced I could cancel it because the step you could cancel was like
2ax + 2bx^2 + abx / (some junk in terms of a and b). Sure ab might be tiny, and many orders of magnitude smaller than a or b alone, but can we really just cancel the term and say “whelp, might as well pretend it doesn’t exist!”?
ETA: It’s not like I saw you could do it by cancelling the term but didn’t because it felt bad. I didn’t even think to do it because cancelling off an arbitrary term just because “it’s small” didn’t even occur to me.
At very least, the teacher should have justified it in a classroom example. By solving a problem on the chalkboard (okay, I’m old) and discarding the minuscule terms that don’t contribute to the “engineering aspect” of the solution, he’d point the way for you to use the same technique on a test.
Shabby teacher who doesn’t teach the tools and techniques needed to solve his test problems!
I had to take multivariable calc for my chemistry major. I somehow got a very high grade, but I think the professor made a mistake. That, or it was his last semester before leaving the school and he just gave everyone A grades, which were few and far between at my school.
It was required for chemistry, but we barely needed any of it for pchem. We used 3D integrals and spherical coordinates, but that’s like lectures #1 and 2 from the math class. The rest was not so useful. They didn’t have us take linear algebra or differential equations. I would have loved a combined “math for chemists” class (physics departments sometimes do this for their majors) that included the essentials of these three courses, plus maybe some data analysis tricks.
But yeah, if only those were the worst of my problems today!
Numbers? Pffft. We don’t use numbers in physics. Well, maybe the odd “2” now and again, or a “3” once in a blue moon. If you have to use numbers, then you’re probably using the wrong units.
But if you want to get rid of some irritating, small factor just call it “epsilon”. Everyone knows anything called “epsilon” ain’t worth shit!
Add chemistry to that list. I took chem 1A and 1B (qualitative and quantitative) at a state university. We had a lecture hall with 100 plus students and a professor who had one speed, fast and furious. Then a lab with a teaching assistant who, if she understood chemistry, hid it well from the lab students.
To the OP. What worked best for me was study groups. You’d be surprised how many people are in your same boat. Together, you’ll get through it though.
I made the OP 8 years ago, I just revived it because of nostalgia for my time in college.
What did help was in newtonian physics I had a great TA who was good at teaching the subject to students one on one. That was what made Physics I so much easier than physics II. I think I got a B+ in newtonian and a C in electrophysics. That is the best thing in my experience if you don’t understand the subject is finding a good TA who can work with you one on one. But finding a good one who has the time is hard.
I’m inclined to agree with you, and to wonder what Carmady found so particularly difficult about multivariable calculus (though admittedly triple integrals can get a bit hairy).
I’m including the preparation of the students and the way the class is taught. When you take multivariable calculus you are almost certainly not prepared for any rigorous explanation of the concepts, and the class is almost certainly not going to provide one. It will likely be the least rigorous class you take. Rather, you have to use very complex calculations and concepts without really understanding them. Green’s Theorem and Stokes’ Theorem stick in my mind as particularly hard. If you try to learn them on your own (from Spivak’s Calculus on Manifolds, maybe) you see that they really are quite difficult topics. Just from this class, you could easily get the impression that math is going to keep getting more and more complicated and less and less understandable.
But the opposite is true. Most other classes (such as algebra, linear algebra, topology) are able to be taught with more rigor from the start. They are actually pretty easy and fun. Linear algebra being easy makes differential equations easier. Moreover, analysis (through big Rudin) gets more fun and makes more sense when you are proving everything. Funnily enough, Stokes’ Theorem was considered an ugly thing to prove even later on.
My general impression on difficulty is that analysis starts off very difficult and gradually gets easier and more fun, while algebra starts off very easy and fun and gradually gets harder and less fun. That’s why multivariable calculus is the hardest class you will see for a very long time. Once you reach algebraic geometry or category theory (which are certainly harder than m.v. calculus) you may regret ever preferring algebra for its easiness. Something I learned to my own sorrow.
Agreed. Green’s theorem kicked my butt. Hardest math class I’ve ever had. Complex Calculus was much easier, even though, at first glance, it might seem vastly harder. Even Differential Equations was easier!
I can appreciate the OP’s viewpoint to an extent. I find calculus to be intrinsically interesting–at times even fascinating–but I never did well in a truly rigorous calculus class. I think I’m simply too careless on the exams, because when it came to the lectures I was usually able to follow the instructor right along.
At least until we got to Integration by Substitution!
The only real problem I have with integral calculus is that I lack the “math sense” needed. I mean, I understand how integration by parts, or u-substitution, or whatever works, but I just lack that intuitive sense of how to transform this gnarly equation into this nice one that can be easily integrated. Especially when trig is involved.
It doesn’t help that they give you a sheet with like 25 “common integration scenarios” to make things “easier”, but it also gives you the problem of trying to figure out if equation 21, 24, or 25 is the one you should be trying to kludge your function into.
It’s lazy, but these days I mostly just abuse Wolfram for integration unless it’s one of those “baby’s first integral” problems that I can do in my head.
Edit: Though the math sense isn’t an innate thing, it can be developed like anything else. I definitely developed one for algebra, I just never really got a chance to develop it for integral calc. It didn’t help that my teacher was pants.
I think that describes me too. The Chain Rule is easy enough to follow when it is introduced in differential calculus, and I can follow its proof, even if I wouldn’t be able to explain it off the top of my head. But in trying to truly understand u-substitution, which is justified by the chain rule, it’s as if I’m trying to look at it inside-out, as it were.
This seems like as good a place as any to post my complain about Discrete Math, which I am taking now. I love algebra, and while I didn;t love calculus, I certainly saw the use of it.
I don’t see ANY use for Discrete Math. I am majoring in IS, why do I need this crap? It’s awful and while my professor is very good, there are times where she might as well be speaking Latin for the good it does me.
Discrete Math is a little like Number Theory: you’ll never, ever actually use it for anything, but it adds a little richness to your comprehension of the mathematical cosmos.
The useful classes, like Numerical Methods and Mathematical Simulations, are good, certainly. Also often fun! But the weird-ass abstract stuff has its own glory. Think of it as similar to having to take General Education humanities classes. Okay, I took Women’s Studies and Composition & Speech. I’m a better person now. College isn’t a trade school; they really do believe in that well-rounded guff.
Wow, I don’t usually hear discrete math as “useless”, if anything it’s usually considered the more down to earth math.
You really don’t see the use of having sets of objects and the ability to combine them, find common elements, and different elements? The utility of graphs and trees – where you can determine how to find the shortest path between two points, or arrange data in a way that it can be accessed quickly? Methods to quickly sort long lists of data? The theory on how to count how many possible ways there are to do something? The idea of formal logics – where you get things like how to tell if a deductive argument is valid, or sound?
Sure, there’s some weird stuff in there, but in general I found discrete math way more practically useful in an every day sense than being able to find the roots of a parabola, or the area under the cross section of a torus.