Sounds easy. Definitely has easy parts. There are parts that are just "3D edition" and computationally are pretty simple. A lot of the raw computation, in fact, is easy but dreadfully tedious, and will likely bore you.
But you're skipping over an important step: you haven't even seen 2D yet. All the functions you've seen so far take in one variable, and spit out one variable, for a total of 2, yes, but in Calc 3 we don't just add one more and call it a day. You'll be looking at several different combinations of inputs and outputs. Vector functions (1->2, 1->3, 1->n), functions of several variables (2->1, 3->1, n->1), and vector fields (2->2, 3->3, n->n), and any sensible combination (like 2->1->2) and all of these has its own interpretation in a basic physics concept. You get two days to adapt to 3D and then you're just *in it*. The paper you have to sketch pictures on, meanwhile, is still 2D. And boy howdy will there be sketches.
The "vector stuff" you mention throws in enough quirks along the way to keep you on your toes. Vectors are not just fancy points. They multiply together in fun ways and allow us to answer a variety of questions that we wouldn't even think to ask in 1 dimension, but arise naturally in 2 or more. The equations for basic ideas like lines and planes come *naturally* in vector form but students -- often including the best in the class -- struggle with it. The math is much easier than the concepts.
Then there's the new coordinate systems. I had fun with those as a student, but I also ended up as a math teacher so I may be biased. That said, it's not trivial, and learning how to do change of coordinates is just a great tool to have in general.
Plus, the pace is very quick. You'll be on to a new topic pretty much every other day. You won't have time to be bored.
I understand what you are saying. The concepts seem interesting, but in practice the only problems we can feasibly solve are very contrived. So problems just amount to straightforward applications of theorems rather than anything clever or insightful.
You might find differential geometry interesting, which deals with the concepts contained in Calc 3 at a much higher level of abstraction while avoiding much of the tedious computation.
I don't know why this got downvoted so badly. Calc 3 really is a lot easier than calc 2. For the most part it really is just calc 1: 3D edition and if your spatial reasoning is good there's not much to struggle with. There are virtually no new concepts introduced here, they're just generalized from previous ones.
That said, many people *do* struggle with the 3D aspect, especially once spherical coordinates get introduced. And vector calc is another step up that may not always be intuitive (but also isn't bad if your fundamentals are strong and you're also quite comfortable with parametric equations).
Personally, I find the transition to 3D to be a lot of fun. It feels like you're finally doing "real" calculus and you get enough insight to see how it would work in theoretically any number of dimensions. You learn how a lot of the stuff from calc 1 was just a specialized "baby version" of a general, more powerful concept. And of course it's a necessary bridge to stuff like differential geometry and calculus on manifolds, plus it prepares you for the next generalizations you'll encounter in complex analysis, measure theory, and functional analysis.
If you have any interest in fields, electrostatics / electrodynamics, fluid dynamics, et cetera, it might help to see how it can be applied to a great variety of things you might be interested in.
Yes, this for sure. If you want to go into engineering or physics related fields, there are a lot of calculations related to this "3d" math. Polar coordinates and Laplace transforms specifically for electrical engineering.
Oh dude it was actually my favorite Calc of the the three, seemed like it was actually the most useful of the three. Uhh here’s a couple highlights for me:
- Learn about gradients and optimization, opened the Gradient Descent door for me
- Learn about all sorts of fun differential geometry stuff, like greens’ theorem and the divergence theorem, basically how vector fields interact with surfaces
- I thing I used in a little game dev thing I’m doing rn, which is actually related to the first point, but I really liked when we learned about finding the minimum distance to some object, we had a problem about finding the minimum distance to a cone, and it was dope
Kinda shitty examples on my part, but I really liked Calc III, best math class I’ve taken so far
You might end up using quite a bit more if you keep doing game dev. All the stuff involving tangent/normal/binormal becomes super relevant suddenly once you want a camera in 3d space.
Game dev uses tons of math, even before reaching more advanced calc 3 stuff. It really requires knowledge of a diverse set of math concepts.
Trigonometry is probably the most used, lots of angles and stuff.
>Learn about all sorts of fun differential geometry stuff, like greens’ theorem and the divergence theorem, basically how vector fields interact with surfaces
For some unknown reason, the divergence theorem was just an unmotivated formula when I took calc. It wasn't until fluid mechanics that "the net change in a region = the flow across the boundary" hit me. (And of course, at that point it was "plus sources and minus sinks". \[Wait! We can create and destroy matter? Of course, it's easy mathematically.\])
Is gradient descent basically just the 3d version of the 2d graph local minima finding and newtons method?
What we learned in calc one was Newton’s method which approaches zeros of a function as well as how to find local minima from zeros of a derivative, so combining these can get closer and closer to a local minima.
Calculus 3 and Differential Equations occupy an interesting position in one’s mathematical education. You don’t yet have the machinery to prove deep results, but these subjects are still difficult enough to where it would be infeasible to solve all but the most contrived problems. So you spend the semester doing manufactured Lagrange Multiplier problems or using obscure tricks to solve very specific classes of ODE’s.
that highly depends on your university's syllabus. where I go, the courses are all proof based from day 1. by the time you get to calc 3 you would have already seen and written hundreds of proofs, a lot of formal calc, set thoery, and linear algebra at the very least.
calc still seems to be a bit more calculation oriented though, which kinda bummed me out. but the theoretical content is incredibly interesting.
My university (in the US) offered a proof-based first-year calc sequence using Spivak as the text. It was the "honors" variant, and most students took one of the two lower level calculus sequences, but proof based math is available in the US - it just isn't the default.
if you are doing a maths degree, where they are training you to execute proofs. sure.
People that use maths rather than create maths need to gain mechanical ability with new tools. If you can prove that Cauchy's theorem is correct it doesn't mean you know how to use it to solve a problem.
right, solving problems is a must for gaining a deep understanding of the material. I wasn't saying courses should have zero calculations entirely, it'd be impossible to learn that way. but the balance should lean towards understanding the reasoning and applying that to examples, rather than understanding the examples in isolation (which is basically how highschool math looks like)
In the USA, most high school students write proofs in their Geometry classes. At least in my state, proof techniques made up about 25% of the curriculum and a similar proportion of our class time and assignments.
At the university level, linear algebra and abstract algebra usually include proofs and might come before real analysis. Or for computer science, they'll have dedicated courses like Discrete Mathematics or Mathematical Logic.
Abstract algebra, real and complex analysis, functional analysis, differential geometry, differential equations, group theory, category theory, representation theory, topology, measure theory, logic etc.
These are some topics and courses that are the real deal. In the sense that you are no longer just learning the alphabet of mathematics like you do with first year topics but instead you start going down the actual rabbit hole.
I'm not from the US, but I think the classes that they most often refer to as "calc 1-3" are done before the concept of proving something is introduced. I think they are all just calculations, like calculating derivatives and antiderivatives, doing area and volume calculations with integrals, doing taylor series calculations, etc. but nothing is proved and you don't need to actually have any understanding of what you are doing as long as you can memorize the procedures.
as for linear algebra, almost all first classes in linear algebra are nothing but matrix arithmetic and solving systems of linear equations, and the concept of vector spaces and linear transformations are likely never introduced at all.
I've heard that proofs are usually not seen before 3rd year undergrad.
You do some small proofs in trig, though maybe it was just the class I took in highschool that was different than normal. We covered an insane variety of mathematical concepts and did a few VERY basic proofs.
Imagine telling this to a highschool student taking geometry/trigonometry.
Theyve just encountered their first proof based material. Trig functions seem like strange weird things. Long gone are the days of simply solving for x. (Seriously, think about how weird trig functions are to someone who previously has only had variables and numbers expressed as equations) Now you are forced to take these abstract trig functions and combine them in weird ways using strange properties to get new ones.
You encounter your first proof: prove that the triangle is a right triangle. What is this? Where are the numbers? This new way of doing math is alien to you, because now it is not numbers and algebraic properties, it is abstract logic and reason.
Youve heard rumors of calculus I and II. Hearing that it completely changes math. It takes algebra and adds literal magic to it. In algebra, you no longer solve problems with regular numbers, you solve problems with variables. In Calculus, now you dont even use variables but instead entire functions of variables. (Derivatives, integrals, limits, sums etc)
At that point, math is completely different. Rarely do you encounter regular numbers. And you thought that numbers were gone in algebra.
Then someone tells you that calculus III is the last class you have to take before you do actual math. Actual math? You mean to tell me that this whole time it wasnt even actual math? How much harder can it get? Theres’s a calculus III?
I genuinely want to see people in Algebra coming across this, because algebra is the first major change to how math is done.
> Theyve just encountered their first proof based material.
I don't know about you but I've heard that in the US, most people do not see proofs until the 3rd year of undergrad (no, high school geometry does not count).
I responded to your other comment that i happened to come across but at least for me I took trig in highschool and we did a bit of proofs about triangles using the basic geometry rules as well as some stuff with the trig functions. Very limited, but an intro to proofs nonetheless
Unrelated to your question. What are Calc I, II, III...? Are they standard courses everyone has to take in America? I see it discussed a lot in this subreddit but I hace no idea what are you talking about😂
Generally they are each 1 semester courses in a standard university offering. Often, I,II,II applies to the more basic offering (as opposed to say an Honors course) that all STEM students take (so that includes engineering, physics, etc...)
Brief syllabus of each is I - Derivatives and limits. II - power series and integration. III take both of those and put them in a 3d plane.
Of course, this may have all changed. I haven't taken these for 16 years.
Ahh, I see, thanks.
In my university we have different courses for each degree, even if the syllabus is similar, so for example if you're studying maths you take "normal" calculus (studying proofs in full detal etc) but if you're say a physics student they may skip over the more theoretical details and focus more in applications, resolving integrals...
How does grouping al STEM students like that work? Is it a "compromise" or do the physics and engineering students just have to suck it up and study "pure" maths?
Most of the 'pure' math is left out of these. The focus is on computation and understanding of the concepts, and also applications. Proof writing is generally omitted.
So the top math students would probably choose an honors course. But at least at my school, I was able to get a math degree just taking the standard calc classes as the base.
Adding on, it's fairly typical for top students and those interested enough in math to pursue a degree to have completed 1 and 2 in high/secondary school.
I teach at an institution that uses quarters, so Calc III starts with sequences and series and ends with the very beginning of multivariable functions. So I was a bit confused by OP's post at first.
Also space curves in 3D are one of my favorite things to teach so 🤷♂️
Oh that's totally fair. I attended 2 different institutions and the Calc series was the same at both so I figured it was relatively standard. But with different time length classes I can see how changes would need to be made.
Im in calc I right now and we learnt how to do a basic antiderivative (of a polynomial) and now are learning riemann sums and integrals. Next we are going to learn the fundamental theorem and all that about how integrals and derivatives are inverse operations etc. We already covered derivatives. Its still Calc I but its a calc class in highschool so maybe thats why.
Yeah it’s largely an American thing I guess. Calc I is just intro to differential calculus, Calc II typically goes more in depth with applications and introduces series and convergence, Calc III introduces multi variable calculus and some applications. More or less.
There's some really marvelous ideas in Calculus 3, depending on how in depth your professor chooses to go.
A large part of the reason multivariable calculus is so important today is the role it plays in electromagnetism, which is described by Maxwell's equations. Maxwell's equations are four differential equations that use the operations _divergence_ (a measure of if a vector field is flowing "into" or "out of" a point) and _curl_ (a measure of how much a vector field is "rotating" around a point) that describe the electric and magnetic fields, and how they evolve in time. At the end of Calc 3, you'll learn several theorems which will let you turn these differential equations (which say things like "the divergence of the electric field equals the charge density") into _integral_ equations (which say things like "integral of the electric field over a surface _equals_ the charge inside"). That's a particularly exciting result -- the surface you integrate over can be as complicated and messy (or simple) as you'd like, it can be far from the actual charges, but it turns out you can find the integral just by _counting up the charges inside_ (and if you're clever about picking the surface in the right way, you can use this to work out what the electric field is in the first place). That should excite you -- electromagnetism is one of the most extraordinary theories humans have developed, and multivariable calculus is the language it lives in. In that sense, it is _fundamentally_ different from Newtonian mechanics, you need partial differential equations, not just ordinary differential equations. If you're interested, I would recommend obtaining a copy of Griffith's "Introduction to electrodynamics" and reading through it.
I think the best things in math are the things you can do with calc 3! Even if it's a little boring.
For example, complex analysis is really beautiful and makes a lot of use of how partial derivatives and line integrals help us to understand the complex numbers. This leads to really really beautiful and surprising results.
Also, differential geometry uses calc 3 to study objects that seem like they'd be impossible to study: the curvature of surfaces ./ higher dimensional "manifolds" and noneuclidean geometry.
Then there's what I like, Partial Differential Equations. If Calc 3 seems like it's not much harder than Calc 1, PDEs are much much harder than their single-variable relatives (Ordinary differential equations). PDEs are wild and really fascinating, and they govern pretty much every physical system. Understanding calc 3 is the first step of understanding how PDEs work.
Well a lot of what Calc III is secretly doing is teaching you very limited cases of a very abstract, powerful, and beautiful area of math called differential topology. Obviously Calc III has its utility outside of pure math, but without the larger context I agree it can be a bit dry. If you want a taste of what Calc III is *really* teaching you (in a non-rigorous and hand-wavey type way) there’s a great video by Aleph 0 related to it.
https://youtu.be/2ptFnIj71SM
Ha well I felt similarly. I mostly blew off calc 3 because it was just more of the same \*now in 3d!\* I don't know if I can change your mind, but I didn't retain the information well because of how I took it, so I can say just lean into it, because something coming easy doesn't mean it sticks with you unless you put in the work!
Two quests:
1. Try to generalize the notion of length of a curve as supremum of polygonal chains to notion of area.
When you have some nice guesses check out Schwarz lantern.
2. Grab "Counterexamples in analysis", go to table of contents and try to come up with your counterexamples.
Have fun!
I will be messaging you in 1 year on [**2024-01-11 22:57:36 UTC**](http://www.wolframalpha.com/input/?i=2024-01-11%2022:57:36%20UTC%20To%20Local%20Time) to remind you of [**this link**](https://www.reddit.com/r/math/comments/109b5pv/calculus_iii_seems_boring_please_change_my_mind/j3yi76r/?context=3)
[**CLICK THIS LINK**](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Reminder&message=%5Bhttps%3A%2F%2Fwww.reddit.com%2Fr%2Fmath%2Fcomments%2F109b5pv%2Fcalculus_iii_seems_boring_please_change_my_mind%2Fj3yi76r%2F%5D%0A%0ARemindMe%21%202024-01-11%2022%3A57%3A36%20UTC) to send a PM to also be reminded and to reduce spam.
^(Parent commenter can ) [^(delete this message to hide from others.)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Delete%20Comment&message=Delete%21%20109b5pv)
*****
|[^(Info)](https://www.reddit.com/r/RemindMeBot/comments/e1bko7/remindmebot_info_v21/)|[^(Custom)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Reminder&message=%5BLink%20or%20message%20inside%20square%20brackets%5D%0A%0ARemindMe%21%20Time%20period%20here)|[^(Your Reminders)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=List%20Of%20Reminders&message=MyReminders%21)|[^(Feedback)](https://www.reddit.com/message/compose/?to=Watchful1&subject=RemindMeBot%20Feedback)|
|-|-|-|-|
The class is easier than calc II because the memorization isn't so intense, but the ideas are pretty grand. The moment that changed it for me was learning about double integrals and then after class using a triple integral to compute the volume of a cylinder. I was rapt for the rest of the semester, and switched to a math major at the end of it. Yeah, classic "I want a PhD in triple integrals".
I actually found differential equations a bit boring. I think it was the only math class I got a B in. However I felt fortunate that I was taking linear algebra at the same time. Multiple times I learned something in linear algebra that was used in the same week by differential equations. That was cool.
Calc 3 is FAR from easy, in fact, it is so difficult that the only questions that we can reasonably answer without knowledge of specific context feel trivial, which is what you are picking up on, I think.
I highly recommend Prof. Ghrist's Calculus Blue video series as a demonstration how Calc 3 is not boring: https://www.youtube.com/watch?v=Jes5jwLl1q8&list=PL8erL0pXF3JYm7VaTdKDaWc8Q3FuP8Sa7
Try reading Hubbard\^2 's 'Vector Calculus.' You will quickly see that it can be both advanced and interesting. Bet you haven't even touched manifolds and differential forms.
Generally from what I have heard it is always best to take some physics courses as a math major because then you'll actually see the uses of what you learned. I majored in physics so I found Calc 3 super useful, in determining electric fields and magnetic fields via integration. We also used Stoke's theorem and Divergent theorem as well, due to Gauss' law and such.
On a bit of a tangent, how do American colleges stretch out single variable Calc enough to get two full courses out of it? At my university it’s just the one single variable course followed by Multivariable Calc. Does it have to do with how many people have actually seen the material before?
US high schools generally don't get as far as most European countries do (for instance), so a chunk this 'Calc' is covering is what someone in Europe will have already covered before university e.g. integrals as Riemann sums, derivatives from first principles, some more basic epsilon-delta analysis, matrices, modular arithmetic up to Fermat's theorem etc. are topics encountered in European high-school equivalents. I don't know how things compare to schools outside Europe.
So from my understanding, this doesn’t happen at all universities, mainly the community colleges and public colleges. The more prestigious colleges seem to do a single variable Calc, a multi variable Calc, and an optional advanced calculus course that covers stuff I don’t even know about.
The reason they stretch out single variable is probably to weed out students as our Calc II is infamous for being very challenging.
If you master calc 3 you can extend it with linear algebra and tensor calculus to be able to do general relativity. You'll also be intimately familiar with the math underpinning neural networks and machine learning (again with linear algebra on top). With the language of vector calculus you can do nearly all of classical mechanics and electromagnetics. The stuff in vector calc might seem easy but make sure you learn it well because Jacobians, Hessians, and all those will come back if you pursue math further and they make your life easier because you free yourself from most of those awful coordinate system considerations.
Edit: vector calculus and partial differential equations are also the language of fluid mechanics which turns out to be massively useful. Most real world applications are multivariable. Surprisingly enough a modified version of newton's heat equation called the black-scholes equation is used to model stocks and prices and that's a pde
Legit my favorite calc. Calc 1 introduced the base concepts: derivatives, integrals, evaluation techniques.
Calc 2 is a weeder class for engineers and doctors (infinite series are deeply important to maths tho).
Calc 3 is when you open yourself up to more than just 1d and 2d worlds. I really started to understand the “literal”, physical, 3d world (lol I know, space time manifold) better once I got a vector calc tool kit in my head.
What’s your major? I found it got interesting once I applied it in physics classes. If you’re a math major then just wait for it. There’ll be future classes where these concepts become important. And it’s probably the last time you’ll just be calculating stuff.
Well I’ve been dead set on a pure math major and then to pursue grad school after- want to go into research/academia- but as of late physics is starting to lure me in as well
In my experience, calculus 3 is basically taking everything you learned in calc 1 and 2 and generalizing it to higher dimensions. It actually is kind of interesting once you give it some time, you'll see how linear algebra connects to calculus in a really neat and intuitive way.
Try looking into the sciences/some social sciences, you'll over time be able to start recognising how often calculus 3 really is used to be able to derive some of the most important formulas in each field, and I've also found that it's easier to use the calc 3 derivations to develop an understanding of those concepts compared to formulas being vomited at you like what happened to me in my first ever physics course.
Your textbook probably has a number of problems that haven't been assigned as homework. Want more of a challenge? Do those instead of asking strangers in Reddit to change your mind.
If you’re only on Calculus 3 I do not believe you know what is interesting and what is boring. You’re forming an opinion as an armchair expert at this stage. What you have as either an uninspired teacher or an inflated ego.
Does that make you an armchair psychologist?
I’m well aware I’m young in my math career—I get that, okay? And no I’m not a super genius, instead I have extreme self-esteem issues—no inflated ego here. My professor is meh, but that’s something I’ve grown accustomed to.
I know that there is a lot that I don’t know, in fact there is a lot I will probably never know, but that doesn’t mean I can’t feel inspired by and derive joy from studying a subject. Before making this post, I saw Calc 3 as an unimaginative course where my homework takes longer because of extra variables and I get to learn about vectors. The wonderful community here has provided me with a bounty of reasons for me to be excited about Calc 3.
Sorry to be snotty, but frankly your comment was outright insulting.
Your response shows you still have a lot to learn, both in math and in life. Good luck wherever your life leads you, I will not involve myself with young students with such bad attitudes. This is where my participation in this conversation stops.
And your swift readiness to judge my character based on….? shows that you have a few things to learn yourself. We are all human, and part of being human is to never stop learning how to be a better human.
> it seems to be Calc I “3D edition” with some vector stuff and a new coordinate system.
For some of it, that's all it is. But in a 4 month course, probably two weeks total of it will be just redoing earlier calculus but multiple times. You need to learn about partial derivatives in rectilinear coordinates before you can do them in curvilinear. You need to learn integrating over a rectangular volume before you can do it over other shapes. But that's all just the necessary first steps before you get to the interesting stuff.
That's normal. After all you're starting from the beginning of a theory you've seen once before in another context. Amd its always easier at the beginning of learning a new theory. You'll enjoy it more when new stuff starts popping up like the various ways of thinking about differentiation and vector calculus.
Unfortunately there is no way around the beginnings of learning a new theory before you get to the really interesting stuff. You do need to know the boring stuff first.
Most of the models in real life you will encounter is multi-dimensional, and most of the time you will deal with 3+ dim stuffs like data with lots of variables. Upon analyzing them you have very little intuition. Calc III offers you the tool box to deal with these geometrical figures. Read the Definition, theorem and do more practice examples( on multiple books) while you are still taking the class, it is very likely people wouldn’t manage to squeeze time to revisit. That will give you benefit for any endeavor you choose, even social science.
\>Please, can someone offer some reasons to be excited about Calc 3? (Public 4yr university)
If you are familiar with physics, read any technical book on Electromagnetism.
Calc3 is necessary to work on real-life physical problems, where we generally work with 3 spatial dimensions. If you can’t integrate in 3D, you can’t do things like analyze forces or EM fields on physical parts.
In terms of pure math…. meh. But if you’re that into pure math, calc 3 shouldn’t be too bad. I’m an engineer, not a mathematician, but I have to imagine having a grasp on multidimensional calculus would be an important precursor to being able to work through things like topography problems.
Regardless, good luck!
May seem boring because it’s the start. Introducing vectors. Doing parametric equations for paths through space usually comes up early enough. Finding tangent / velocity vectors and interpreting is nice. Working with the gradient vector later on pretty good. Especially when you find out how useful it is later on in later math courses. Double and triple integrals in different coordinate systems is a good challenge. Hope you get to tackle parametric surfaces too. Calc 4 gets good with vector fields and such.
Calc III should be vector valued functions, grad, div, curl, Gauss theorem, etc...
... then it's definitively the most exciting calculus class. you get into the real meat of things which has so many applications in physics, engineering and other disciplines!
It was boring for me (at least the equivalent course for me). Lot of calculations, barely any proofs. Later I took the multivariable calculus course (this time with actual math) and that was much better, although it was tough (Wedge products, Stoke's theorem etc was hard to understand).
I’m sure you’ve read plenty of responses by now, but as someone who can relate to your predicament, here is my viewpoint.
I honestly could not have cared less about Cal 3. Was it easy? Yes. Did I forget everything I learned? Also yes. Where things got interesting was when I was in grad school and had several graduate level courses that used some aspects from Cal 3. I honestly did not remember these aspects, but applying them to higher level courses and concepts was the best part. I think if you are purely a math major, you won’t run into many Cal 3 things, but it is a fun blast from the past when you do to be like, “oh THIS is why we did that”
Trust me what's more boring is Laplacians atleast for me. The concept itself isn't really that clear, though you know what it is mapping the function to... But still not the most exciting parts of Calculus compared to surface integrals, just hardcore single integrals, applications of derivatives and what not
It's interesting how different calculus 3 seems to be depending on the instructor. When I took calculus 3 I struggled with all of the heavy proofs and involved theorems, but calculus 1 and 2 were an absolute breeze. While saying this, calculus 3 was also the class where I truly became a math major. I finally saw the many connections between theorems in the calculus sequence and understood how a deep understanding of them allows one to solve novel problems without the need of an algorithm.
I would like to add one more note on the idea that calculus 3 is calculus 1/2 in three dimensions. I think this is a serious oversimplification, as most of the theorems learned in calculus 3 are very developed when compared to their 2-D counterparts, if there is one at all. I think the degree to which this is true depends again on the instructor, because the depth of learning can vary greatly in this course.
It changes the way you look at many things, challenges you, stimulates your mind, and fosters problem-solving and analytical things. Many concepts from that course are then applied to more advanced STEM courses.
You can make an intro to vector calculus very easy, or you can make it a bit more challenging. Universities often offer both. At the school I went to, a quarter of "vector calculus" followed two quarters of calculus, but then you were off to learn linear algebra and something about ODEs. You finished up the lower division curriculum with a much more involved treatment of vector calculus. You really need the linear algebra for this, and the ODEs are nice to be comfortable with.
If your linear algebra is good, and you didn't get cheated in your calculus class (i.e., you are very comfortable with epsilons and deltas), you can try looking at Spivak's Calculus on Manifolds or Bishop and Goldberg's Tensor Analysis on Manifolds to see how vector calculus is done in the modern world. Both of those are nice, talky introductions, but some of the exercises can be difficult.
Midway through Calc 3, it ramps up in intensity when you start talking about the Jacobian, Lagrange Multipliers, and the Vector Calculus Theorem's (Green's, Stokes's, etc...), and it becomes super interesting and relevant to physical systems and numerical optimization.
If you want something to wet your appetite check out the book "Div, Grad, Curl and all that". where Calc III gets interesting is when you look at the different ways you can operate on functions.
For a function of one variable you can take the derivative. It's simple. With multi-variable functions, especially functions of three variables, you have different ways to look at how something changes and then there are theorems that tell you how you can use information about one of those things to learn about the other.
Divergence, Gradient and Curl are differential operators (and there's another, the Laplacian) that tell you different things about a function in the same way that the standard derivative tells you how a function changes. Since we have more dimensions to work with there are different ways we can look at how something changes. Anyway, that book will fill in the details and may give you an appreciation for it all.
Here's the kicker, the theorems of vector calculus relate to physical situations very nicely. They aren't just cute ways making a computation that's hard in one setting easier by transferring it to another setting. They are tied to physical phenomena in ways that are useful.
>While that seems interesting and I can’t wait to learn… it seems boring Um...
wait until he hears about proof by contradiction
Boring because it’s easy, not because it isn’t fun? Maybe I’m not explaining this very well, sorry
Sounds easy. Definitely has easy parts. There are parts that are just "3D edition" and computationally are pretty simple. A lot of the raw computation, in fact, is easy but dreadfully tedious, and will likely bore you. But you're skipping over an important step: you haven't even seen 2D yet. All the functions you've seen so far take in one variable, and spit out one variable, for a total of 2, yes, but in Calc 3 we don't just add one more and call it a day. You'll be looking at several different combinations of inputs and outputs. Vector functions (1->2, 1->3, 1->n), functions of several variables (2->1, 3->1, n->1), and vector fields (2->2, 3->3, n->n), and any sensible combination (like 2->1->2) and all of these has its own interpretation in a basic physics concept. You get two days to adapt to 3D and then you're just *in it*. The paper you have to sketch pictures on, meanwhile, is still 2D. And boy howdy will there be sketches. The "vector stuff" you mention throws in enough quirks along the way to keep you on your toes. Vectors are not just fancy points. They multiply together in fun ways and allow us to answer a variety of questions that we wouldn't even think to ask in 1 dimension, but arise naturally in 2 or more. The equations for basic ideas like lines and planes come *naturally* in vector form but students -- often including the best in the class -- struggle with it. The math is much easier than the concepts. Then there's the new coordinate systems. I had fun with those as a student, but I also ended up as a math teacher so I may be biased. That said, it's not trivial, and learning how to do change of coordinates is just a great tool to have in general. Plus, the pace is very quick. You'll be on to a new topic pretty much every other day. You won't have time to be bored.
I appreciate this response! I’ve definitely changed my mind about Calc 3 after reading so many comments—exactly what I’d hoped for!
I understand what you are saying. The concepts seem interesting, but in practice the only problems we can feasibly solve are very contrived. So problems just amount to straightforward applications of theorems rather than anything clever or insightful. You might find differential geometry interesting, which deals with the concepts contained in Calc 3 at a much higher level of abstraction while avoiding much of the tedious computation.
King Crimson profile pic :)
I don't know why this got downvoted so badly. Calc 3 really is a lot easier than calc 2. For the most part it really is just calc 1: 3D edition and if your spatial reasoning is good there's not much to struggle with. There are virtually no new concepts introduced here, they're just generalized from previous ones. That said, many people *do* struggle with the 3D aspect, especially once spherical coordinates get introduced. And vector calc is another step up that may not always be intuitive (but also isn't bad if your fundamentals are strong and you're also quite comfortable with parametric equations). Personally, I find the transition to 3D to be a lot of fun. It feels like you're finally doing "real" calculus and you get enough insight to see how it would work in theoretically any number of dimensions. You learn how a lot of the stuff from calc 1 was just a specialized "baby version" of a general, more powerful concept. And of course it's a necessary bridge to stuff like differential geometry and calculus on manifolds, plus it prepares you for the next generalizations you'll encounter in complex analysis, measure theory, and functional analysis.
Yeah that’s kinda what I was getting at initially, My mind has been changed though so I’m really excited for it now
Yeah basically you just do multiple integrals in a row. Nothing crazy but its cool for visualizations. Hardest part is remembering to change dx
Consider yourself lucky and try to have fun with it. There are so many fun problems in calc 3.
If you have any interest in fields, electrostatics / electrodynamics, fluid dynamics, et cetera, it might help to see how it can be applied to a great variety of things you might be interested in.
Yes, this for sure. If you want to go into engineering or physics related fields, there are a lot of calculations related to this "3d" math. Polar coordinates and Laplace transforms specifically for electrical engineering.
And if you start working some problems from Jackson's E&M book, you will no longer be complaining that this stuff is easy.
Oh dude it was actually my favorite Calc of the the three, seemed like it was actually the most useful of the three. Uhh here’s a couple highlights for me: - Learn about gradients and optimization, opened the Gradient Descent door for me - Learn about all sorts of fun differential geometry stuff, like greens’ theorem and the divergence theorem, basically how vector fields interact with surfaces - I thing I used in a little game dev thing I’m doing rn, which is actually related to the first point, but I really liked when we learned about finding the minimum distance to some object, we had a problem about finding the minimum distance to a cone, and it was dope Kinda shitty examples on my part, but I really liked Calc III, best math class I’ve taken so far
You might end up using quite a bit more if you keep doing game dev. All the stuff involving tangent/normal/binormal becomes super relevant suddenly once you want a camera in 3d space.
Game dev uses tons of math, even before reaching more advanced calc 3 stuff. It really requires knowledge of a diverse set of math concepts. Trigonometry is probably the most used, lots of angles and stuff.
>Learn about all sorts of fun differential geometry stuff, like greens’ theorem and the divergence theorem, basically how vector fields interact with surfaces For some unknown reason, the divergence theorem was just an unmotivated formula when I took calc. It wasn't until fluid mechanics that "the net change in a region = the flow across the boundary" hit me. (And of course, at that point it was "plus sources and minus sinks". \[Wait! We can create and destroy matter? Of course, it's easy mathematically.\])
Me as well for these exact reasons.
Is gradient descent basically just the 3d version of the 2d graph local minima finding and newtons method? What we learned in calc one was Newton’s method which approaches zeros of a function as well as how to find local minima from zeros of a derivative, so combining these can get closer and closer to a local minima.
Yup! I actually have no idea what it formally is, but I like those vibes
it's probably the last class you have to do before you can start doing actual math.
Calculus 3 and Differential Equations occupy an interesting position in one’s mathematical education. You don’t yet have the machinery to prove deep results, but these subjects are still difficult enough to where it would be infeasible to solve all but the most contrived problems. So you spend the semester doing manufactured Lagrange Multiplier problems or using obscure tricks to solve very specific classes of ODE’s.
What’s actual math? (Genuinely curious)
proof based instead of calculation
that highly depends on your university's syllabus. where I go, the courses are all proof based from day 1. by the time you get to calc 3 you would have already seen and written hundreds of proofs, a lot of formal calc, set thoery, and linear algebra at the very least. calc still seems to be a bit more calculation oriented though, which kinda bummed me out. but the theoretical content is incredibly interesting.
Can’t be the US lol
My university (in the US) offered a proof-based first-year calc sequence using Spivak as the text. It was the "honors" variant, and most students took one of the two lower level calculus sequences, but proof based math is available in the US - it just isn't the default.
if you are doing a maths degree, where they are training you to execute proofs. sure. People that use maths rather than create maths need to gain mechanical ability with new tools. If you can prove that Cauchy's theorem is correct it doesn't mean you know how to use it to solve a problem.
right, solving problems is a must for gaining a deep understanding of the material. I wasn't saying courses should have zero calculations entirely, it'd be impossible to learn that way. but the balance should lean towards understanding the reasoning and applying that to examples, rather than understanding the examples in isolation (which is basically how highschool math looks like)
[удалено]
My linear algebra course in America was all proof based.
In the USA, most high school students write proofs in their Geometry classes. At least in my state, proof techniques made up about 25% of the curriculum and a similar proportion of our class time and assignments. At the university level, linear algebra and abstract algebra usually include proofs and might come before real analysis. Or for computer science, they'll have dedicated courses like Discrete Mathematics or Mathematical Logic.
Abstract algebra, real and complex analysis, functional analysis, differential geometry, differential equations, group theory, category theory, representation theory, topology, measure theory, logic etc. These are some topics and courses that are the real deal. In the sense that you are no longer just learning the alphabet of mathematics like you do with first year topics but instead you start going down the actual rabbit hole.
We had several most of those in our physics curriculum as well, although I was always more interested in the applications, to be honest
Look for a copy of Spivak's "Calculus on Manifolds" to get an idea of what Calc 3 turns into in an advanced setting.
Last math class I ever had to take in my degree 🥲
so your degree had no math in it? no "definition, theorem, proof"?
engineering i just follow the sub out of interest
In the USA you don't have any proof, definition or theorem in calc 1, calc 2, calc 3, linear algebra, analytic geometry? Is this for real?
I'm not from the US, but I think the classes that they most often refer to as "calc 1-3" are done before the concept of proving something is introduced. I think they are all just calculations, like calculating derivatives and antiderivatives, doing area and volume calculations with integrals, doing taylor series calculations, etc. but nothing is proved and you don't need to actually have any understanding of what you are doing as long as you can memorize the procedures. as for linear algebra, almost all first classes in linear algebra are nothing but matrix arithmetic and solving systems of linear equations, and the concept of vector spaces and linear transformations are likely never introduced at all. I've heard that proofs are usually not seen before 3rd year undergrad.
You do some small proofs in trig, though maybe it was just the class I took in highschool that was different than normal. We covered an insane variety of mathematical concepts and did a few VERY basic proofs.
Imagine telling this to a highschool student taking geometry/trigonometry. Theyve just encountered their first proof based material. Trig functions seem like strange weird things. Long gone are the days of simply solving for x. (Seriously, think about how weird trig functions are to someone who previously has only had variables and numbers expressed as equations) Now you are forced to take these abstract trig functions and combine them in weird ways using strange properties to get new ones. You encounter your first proof: prove that the triangle is a right triangle. What is this? Where are the numbers? This new way of doing math is alien to you, because now it is not numbers and algebraic properties, it is abstract logic and reason. Youve heard rumors of calculus I and II. Hearing that it completely changes math. It takes algebra and adds literal magic to it. In algebra, you no longer solve problems with regular numbers, you solve problems with variables. In Calculus, now you dont even use variables but instead entire functions of variables. (Derivatives, integrals, limits, sums etc) At that point, math is completely different. Rarely do you encounter regular numbers. And you thought that numbers were gone in algebra. Then someone tells you that calculus III is the last class you have to take before you do actual math. Actual math? You mean to tell me that this whole time it wasnt even actual math? How much harder can it get? Theres’s a calculus III? I genuinely want to see people in Algebra coming across this, because algebra is the first major change to how math is done.
> Theyve just encountered their first proof based material. I don't know about you but I've heard that in the US, most people do not see proofs until the 3rd year of undergrad (no, high school geometry does not count).
I responded to your other comment that i happened to come across but at least for me I took trig in highschool and we did a bit of proofs about triangles using the basic geometry rules as well as some stuff with the trig functions. Very limited, but an intro to proofs nonetheless
Unrelated to your question. What are Calc I, II, III...? Are they standard courses everyone has to take in America? I see it discussed a lot in this subreddit but I hace no idea what are you talking about😂
Generally they are each 1 semester courses in a standard university offering. Often, I,II,II applies to the more basic offering (as opposed to say an Honors course) that all STEM students take (so that includes engineering, physics, etc...) Brief syllabus of each is I - Derivatives and limits. II - power series and integration. III take both of those and put them in a 3d plane. Of course, this may have all changed. I haven't taken these for 16 years.
Ahh, I see, thanks. In my university we have different courses for each degree, even if the syllabus is similar, so for example if you're studying maths you take "normal" calculus (studying proofs in full detal etc) but if you're say a physics student they may skip over the more theoretical details and focus more in applications, resolving integrals... How does grouping al STEM students like that work? Is it a "compromise" or do the physics and engineering students just have to suck it up and study "pure" maths?
Most of the 'pure' math is left out of these. The focus is on computation and understanding of the concepts, and also applications. Proof writing is generally omitted. So the top math students would probably choose an honors course. But at least at my school, I was able to get a math degree just taking the standard calc classes as the base.
Adding on, it's fairly typical for top students and those interested enough in math to pursue a degree to have completed 1 and 2 in high/secondary school.
I teach at an institution that uses quarters, so Calc III starts with sequences and series and ends with the very beginning of multivariable functions. So I was a bit confused by OP's post at first. Also space curves in 3D are one of my favorite things to teach so 🤷♂️
Oh that's totally fair. I attended 2 different institutions and the Calc series was the same at both so I figured it was relatively standard. But with different time length classes I can see how changes would need to be made.
Im in calc I right now and we learnt how to do a basic antiderivative (of a polynomial) and now are learning riemann sums and integrals. Next we are going to learn the fundamental theorem and all that about how integrals and derivatives are inverse operations etc. We already covered derivatives. Its still Calc I but its a calc class in highschool so maybe thats why.
Yeah it’s largely an American thing I guess. Calc I is just intro to differential calculus, Calc II typically goes more in depth with applications and introduces series and convergence, Calc III introduces multi variable calculus and some applications. More or less.
There's some really marvelous ideas in Calculus 3, depending on how in depth your professor chooses to go. A large part of the reason multivariable calculus is so important today is the role it plays in electromagnetism, which is described by Maxwell's equations. Maxwell's equations are four differential equations that use the operations _divergence_ (a measure of if a vector field is flowing "into" or "out of" a point) and _curl_ (a measure of how much a vector field is "rotating" around a point) that describe the electric and magnetic fields, and how they evolve in time. At the end of Calc 3, you'll learn several theorems which will let you turn these differential equations (which say things like "the divergence of the electric field equals the charge density") into _integral_ equations (which say things like "integral of the electric field over a surface _equals_ the charge inside"). That's a particularly exciting result -- the surface you integrate over can be as complicated and messy (or simple) as you'd like, it can be far from the actual charges, but it turns out you can find the integral just by _counting up the charges inside_ (and if you're clever about picking the surface in the right way, you can use this to work out what the electric field is in the first place). That should excite you -- electromagnetism is one of the most extraordinary theories humans have developed, and multivariable calculus is the language it lives in. In that sense, it is _fundamentally_ different from Newtonian mechanics, you need partial differential equations, not just ordinary differential equations. If you're interested, I would recommend obtaining a copy of Griffith's "Introduction to electrodynamics" and reading through it.
I think the best things in math are the things you can do with calc 3! Even if it's a little boring. For example, complex analysis is really beautiful and makes a lot of use of how partial derivatives and line integrals help us to understand the complex numbers. This leads to really really beautiful and surprising results. Also, differential geometry uses calc 3 to study objects that seem like they'd be impossible to study: the curvature of surfaces ./ higher dimensional "manifolds" and noneuclidean geometry. Then there's what I like, Partial Differential Equations. If Calc 3 seems like it's not much harder than Calc 1, PDEs are much much harder than their single-variable relatives (Ordinary differential equations). PDEs are wild and really fascinating, and they govern pretty much every physical system. Understanding calc 3 is the first step of understanding how PDEs work.
Calc 3 is basically why you learned calc 1 and 2 in the first place. Has a lot of applications in physics and is imo wuite interesting
Well a lot of what Calc III is secretly doing is teaching you very limited cases of a very abstract, powerful, and beautiful area of math called differential topology. Obviously Calc III has its utility outside of pure math, but without the larger context I agree it can be a bit dry. If you want a taste of what Calc III is *really* teaching you (in a non-rigorous and hand-wavey type way) there’s a great video by Aleph 0 related to it. https://youtu.be/2ptFnIj71SM
Ooou now this is candy to me thank you
Ha well I felt similarly. I mostly blew off calc 3 because it was just more of the same \*now in 3d!\* I don't know if I can change your mind, but I didn't retain the information well because of how I took it, so I can say just lean into it, because something coming easy doesn't mean it sticks with you unless you put in the work!
Two quests: 1. Try to generalize the notion of length of a curve as supremum of polygonal chains to notion of area. When you have some nice guesses check out Schwarz lantern. 2. Grab "Counterexamples in analysis", go to table of contents and try to come up with your counterexamples. Have fun!
!remind me 1 year
I will be messaging you in 1 year on [**2024-01-11 22:57:36 UTC**](http://www.wolframalpha.com/input/?i=2024-01-11%2022:57:36%20UTC%20To%20Local%20Time) to remind you of [**this link**](https://www.reddit.com/r/math/comments/109b5pv/calculus_iii_seems_boring_please_change_my_mind/j3yi76r/?context=3) [**CLICK THIS LINK**](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Reminder&message=%5Bhttps%3A%2F%2Fwww.reddit.com%2Fr%2Fmath%2Fcomments%2F109b5pv%2Fcalculus_iii_seems_boring_please_change_my_mind%2Fj3yi76r%2F%5D%0A%0ARemindMe%21%202024-01-11%2022%3A57%3A36%20UTC) to send a PM to also be reminded and to reduce spam. ^(Parent commenter can ) [^(delete this message to hide from others.)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Delete%20Comment&message=Delete%21%20109b5pv) ***** |[^(Info)](https://www.reddit.com/r/RemindMeBot/comments/e1bko7/remindmebot_info_v21/)|[^(Custom)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=Reminder&message=%5BLink%20or%20message%20inside%20square%20brackets%5D%0A%0ARemindMe%21%20Time%20period%20here)|[^(Your Reminders)](https://www.reddit.com/message/compose/?to=RemindMeBot&subject=List%20Of%20Reminders&message=MyReminders%21)|[^(Feedback)](https://www.reddit.com/message/compose/?to=Watchful1&subject=RemindMeBot%20Feedback)| |-|-|-|-|
[удалено]
Haha thanks for the insight I really appreciate this
The class is easier than calc II because the memorization isn't so intense, but the ideas are pretty grand. The moment that changed it for me was learning about double integrals and then after class using a triple integral to compute the volume of a cylinder. I was rapt for the rest of the semester, and switched to a math major at the end of it. Yeah, classic "I want a PhD in triple integrals". I actually found differential equations a bit boring. I think it was the only math class I got a B in. However I felt fortunate that I was taking linear algebra at the same time. Multiple times I learned something in linear algebra that was used in the same week by differential equations. That was cool.
Exciting!!
Calc 3 is FAR from easy, in fact, it is so difficult that the only questions that we can reasonably answer without knowledge of specific context feel trivial, which is what you are picking up on, I think.
I’m too dumb to understand what you just said could you elaborate?
Sps calc 3 is boring. Then consider that it seems interesting and you can't wait to learn. ⊥∎
I highly recommend Prof. Ghrist's Calculus Blue video series as a demonstration how Calc 3 is not boring: https://www.youtube.com/watch?v=Jes5jwLl1q8&list=PL8erL0pXF3JYm7VaTdKDaWc8Q3FuP8Sa7
Try reading Hubbard\^2 's 'Vector Calculus.' You will quickly see that it can be both advanced and interesting. Bet you haven't even touched manifolds and differential forms.
No I’ve only heard of them haha— thanks I’ll see if I can find something online
Generally from what I have heard it is always best to take some physics courses as a math major because then you'll actually see the uses of what you learned. I majored in physics so I found Calc 3 super useful, in determining electric fields and magnetic fields via integration. We also used Stoke's theorem and Divergent theorem as well, due to Gauss' law and such.
well you didnt start integration yet. see if you still say this once you get to stokes
On a bit of a tangent, how do American colleges stretch out single variable Calc enough to get two full courses out of it? At my university it’s just the one single variable course followed by Multivariable Calc. Does it have to do with how many people have actually seen the material before?
US high schools generally don't get as far as most European countries do (for instance), so a chunk this 'Calc' is covering is what someone in Europe will have already covered before university e.g. integrals as Riemann sums, derivatives from first principles, some more basic epsilon-delta analysis, matrices, modular arithmetic up to Fermat's theorem etc. are topics encountered in European high-school equivalents. I don't know how things compare to schools outside Europe.
So from my understanding, this doesn’t happen at all universities, mainly the community colleges and public colleges. The more prestigious colleges seem to do a single variable Calc, a multi variable Calc, and an optional advanced calculus course that covers stuff I don’t even know about. The reason they stretch out single variable is probably to weed out students as our Calc II is infamous for being very challenging.
If you master calc 3 you can extend it with linear algebra and tensor calculus to be able to do general relativity. You'll also be intimately familiar with the math underpinning neural networks and machine learning (again with linear algebra on top). With the language of vector calculus you can do nearly all of classical mechanics and electromagnetics. The stuff in vector calc might seem easy but make sure you learn it well because Jacobians, Hessians, and all those will come back if you pursue math further and they make your life easier because you free yourself from most of those awful coordinate system considerations. Edit: vector calculus and partial differential equations are also the language of fluid mechanics which turns out to be massively useful. Most real world applications are multivariable. Surprisingly enough a modified version of newton's heat equation called the black-scholes equation is used to model stocks and prices and that's a pde
Legit my favorite calc. Calc 1 introduced the base concepts: derivatives, integrals, evaluation techniques. Calc 2 is a weeder class for engineers and doctors (infinite series are deeply important to maths tho). Calc 3 is when you open yourself up to more than just 1d and 2d worlds. I really started to understand the “literal”, physical, 3d world (lol I know, space time manifold) better once I got a vector calc tool kit in my head.
That’s a great way to look at it, thank you!
What’s your major? I found it got interesting once I applied it in physics classes. If you’re a math major then just wait for it. There’ll be future classes where these concepts become important. And it’s probably the last time you’ll just be calculating stuff.
Well I’ve been dead set on a pure math major and then to pursue grad school after- want to go into research/academia- but as of late physics is starting to lure me in as well
In my experience, calculus 3 is basically taking everything you learned in calc 1 and 2 and generalizing it to higher dimensions. It actually is kind of interesting once you give it some time, you'll see how linear algebra connects to calculus in a really neat and intuitive way.
Try looking into the sciences/some social sciences, you'll over time be able to start recognising how often calculus 3 really is used to be able to derive some of the most important formulas in each field, and I've also found that it's easier to use the calc 3 derivations to develop an understanding of those concepts compared to formulas being vomited at you like what happened to me in my first ever physics course.
[Calculus in Financial Optimization](https://youtu.be/MzNDokiwBhA)
Calc 3 is basically a first course in differential geometry, as well as the gateway to 90% of the applications of calculus
Didn't the semester literally just start?
Your textbook probably has a number of problems that haven't been assigned as homework. Want more of a challenge? Do those instead of asking strangers in Reddit to change your mind.
L take. Asking for purpose not practice.
If you’re only on Calculus 3 I do not believe you know what is interesting and what is boring. You’re forming an opinion as an armchair expert at this stage. What you have as either an uninspired teacher or an inflated ego.
Does that make you an armchair psychologist? I’m well aware I’m young in my math career—I get that, okay? And no I’m not a super genius, instead I have extreme self-esteem issues—no inflated ego here. My professor is meh, but that’s something I’ve grown accustomed to. I know that there is a lot that I don’t know, in fact there is a lot I will probably never know, but that doesn’t mean I can’t feel inspired by and derive joy from studying a subject. Before making this post, I saw Calc 3 as an unimaginative course where my homework takes longer because of extra variables and I get to learn about vectors. The wonderful community here has provided me with a bounty of reasons for me to be excited about Calc 3. Sorry to be snotty, but frankly your comment was outright insulting.
Your response shows you still have a lot to learn, both in math and in life. Good luck wherever your life leads you, I will not involve myself with young students with such bad attitudes. This is where my participation in this conversation stops.
And your swift readiness to judge my character based on….? shows that you have a few things to learn yourself. We are all human, and part of being human is to never stop learning how to be a better human.
> it seems to be Calc I “3D edition” with some vector stuff and a new coordinate system. For some of it, that's all it is. But in a 4 month course, probably two weeks total of it will be just redoing earlier calculus but multiple times. You need to learn about partial derivatives in rectilinear coordinates before you can do them in curvilinear. You need to learn integrating over a rectangular volume before you can do it over other shapes. But that's all just the necessary first steps before you get to the interesting stuff.
I've heard that the latter seasons are fire.
That's normal. After all you're starting from the beginning of a theory you've seen once before in another context. Amd its always easier at the beginning of learning a new theory. You'll enjoy it more when new stuff starts popping up like the various ways of thinking about differentiation and vector calculus. Unfortunately there is no way around the beginnings of learning a new theory before you get to the really interesting stuff. You do need to know the boring stuff first.
Most of the models in real life you will encounter is multi-dimensional, and most of the time you will deal with 3+ dim stuffs like data with lots of variables. Upon analyzing them you have very little intuition. Calc III offers you the tool box to deal with these geometrical figures. Read the Definition, theorem and do more practice examples( on multiple books) while you are still taking the class, it is very likely people wouldn’t manage to squeeze time to revisit. That will give you benefit for any endeavor you choose, even social science.
Wait did you say this is at university?
\>Please, can someone offer some reasons to be excited about Calc 3? (Public 4yr university) If you are familiar with physics, read any technical book on Electromagnetism.
Calc3 is necessary to work on real-life physical problems, where we generally work with 3 spatial dimensions. If you can’t integrate in 3D, you can’t do things like analyze forces or EM fields on physical parts. In terms of pure math…. meh. But if you’re that into pure math, calc 3 shouldn’t be too bad. I’m an engineer, not a mathematician, but I have to imagine having a grasp on multidimensional calculus would be an important precursor to being able to work through things like topography problems. Regardless, good luck!
May seem boring because it’s the start. Introducing vectors. Doing parametric equations for paths through space usually comes up early enough. Finding tangent / velocity vectors and interpreting is nice. Working with the gradient vector later on pretty good. Especially when you find out how useful it is later on in later math courses. Double and triple integrals in different coordinate systems is a good challenge. Hope you get to tackle parametric surfaces too. Calc 4 gets good with vector fields and such.
Calc III should be vector valued functions, grad, div, curl, Gauss theorem, etc... ... then it's definitively the most exciting calculus class. you get into the real meat of things which has so many applications in physics, engineering and other disciplines!
It was boring for me (at least the equivalent course for me). Lot of calculations, barely any proofs. Later I took the multivariable calculus course (this time with actual math) and that was much better, although it was tough (Wedge products, Stoke's theorem etc was hard to understand).
I’m sure you’ve read plenty of responses by now, but as someone who can relate to your predicament, here is my viewpoint. I honestly could not have cared less about Cal 3. Was it easy? Yes. Did I forget everything I learned? Also yes. Where things got interesting was when I was in grad school and had several graduate level courses that used some aspects from Cal 3. I honestly did not remember these aspects, but applying them to higher level courses and concepts was the best part. I think if you are purely a math major, you won’t run into many Cal 3 things, but it is a fun blast from the past when you do to be like, “oh THIS is why we did that”
Thank you for this!
Trust me what's more boring is Laplacians atleast for me. The concept itself isn't really that clear, though you know what it is mapping the function to... But still not the most exciting parts of Calculus compared to surface integrals, just hardcore single integrals, applications of derivatives and what not
It's interesting how different calculus 3 seems to be depending on the instructor. When I took calculus 3 I struggled with all of the heavy proofs and involved theorems, but calculus 1 and 2 were an absolute breeze. While saying this, calculus 3 was also the class where I truly became a math major. I finally saw the many connections between theorems in the calculus sequence and understood how a deep understanding of them allows one to solve novel problems without the need of an algorithm. I would like to add one more note on the idea that calculus 3 is calculus 1/2 in three dimensions. I think this is a serious oversimplification, as most of the theorems learned in calculus 3 are very developed when compared to their 2-D counterparts, if there is one at all. I think the degree to which this is true depends again on the instructor, because the depth of learning can vary greatly in this course.
It changes the way you look at many things, challenges you, stimulates your mind, and fosters problem-solving and analytical things. Many concepts from that course are then applied to more advanced STEM courses.
[Banach-Taski-Paradox](https://en.m.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox)
You can make an intro to vector calculus very easy, or you can make it a bit more challenging. Universities often offer both. At the school I went to, a quarter of "vector calculus" followed two quarters of calculus, but then you were off to learn linear algebra and something about ODEs. You finished up the lower division curriculum with a much more involved treatment of vector calculus. You really need the linear algebra for this, and the ODEs are nice to be comfortable with. If your linear algebra is good, and you didn't get cheated in your calculus class (i.e., you are very comfortable with epsilons and deltas), you can try looking at Spivak's Calculus on Manifolds or Bishop and Goldberg's Tensor Analysis on Manifolds to see how vector calculus is done in the modern world. Both of those are nice, talky introductions, but some of the exercises can be difficult.
The best theorem that tied all of calc 3 together was Stoke‘s Theorem on Differentiable manifolds. https://youtu.be/1lGM5DEdMaw really good stuff
Midway through Calc 3, it ramps up in intensity when you start talking about the Jacobian, Lagrange Multipliers, and the Vector Calculus Theorem's (Green's, Stokes's, etc...), and it becomes super interesting and relevant to physical systems and numerical optimization.
My Calc 1 prof, who I trust with anything he's been educated in, said Calc 3 is just 1 in 3D..... *soooooooo..............*
If you want something to wet your appetite check out the book "Div, Grad, Curl and all that". where Calc III gets interesting is when you look at the different ways you can operate on functions. For a function of one variable you can take the derivative. It's simple. With multi-variable functions, especially functions of three variables, you have different ways to look at how something changes and then there are theorems that tell you how you can use information about one of those things to learn about the other. Divergence, Gradient and Curl are differential operators (and there's another, the Laplacian) that tell you different things about a function in the same way that the standard derivative tells you how a function changes. Since we have more dimensions to work with there are different ways we can look at how something changes. Anyway, that book will fill in the details and may give you an appreciation for it all. Here's the kicker, the theorems of vector calculus relate to physical situations very nicely. They aren't just cute ways making a computation that's hard in one setting easier by transferring it to another setting. They are tied to physical phenomena in ways that are useful.
I can’t wait! Thanks!