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Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in whay fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I'm a 3rd year undergraduate, majoring in pure mathematics. I've done well in the "proof-based" subjects I've taken, and I think what is equivalent equations in math because Equatoins understand the "rules of the game.
This obviously really helps. Recently, however, I've been finding that "technique-based" as opposed to "proof-based" subjects like complex analysis, vector calculus, differential equations etc. It's like when I'm sitting in these lectures, the "logic" of math suddenly becomes opaque. I can never tell what the premises are. I often don't know whether we're trying to show that "A implies B", or whether we're trying to show that "A iff B". Stuff is happening on the board, but the "rules of the game" just aren't clear to me.
Does anyone else have a similar problem with "technique-based" math? And if so, what can be done about it? Let me give an example. Below, I've copied some of this problem from Wikipedia, and I have inserted my own thoughts in italics. Are you asserting that 2 follows from 1or are you saying they're logically equivalent? And I still don't know whether equation 1 is a premise, or what our premises are.
But what's all this "we must check" nonsense? If the argument was just laid out in a coherent fashion, nonsense like "we must check" simply wouldn't appear. Then why don't you just say so? I honestly can't tell. Well you get the general gist. So my question is, can other people relate to this, and what can be done about it? I can actually relate fairly well to this. Please don't take this the wrong way, but my main advice would be:.
By this I don't mean that you should whay be concerned about your grades getting worse. But while it appears that you have trained yourself to solid, rigorous thinking in proof-making and such, you have not yet learned to simply apply techniques, and worry later. You need to learn to:. I hope this doesn't strike you as a silly self-help program, and a pat on the back that all will be well.
This isn't easy, and, when much younger, I found myself in rather a similar situation. In my math undergrad studies, everything was proofs. And then I studied in France and was blown away by the skill of the average student to simply get stuff done, and to do it fast, matn well I had similar experiences with theoretical what is equivalent equations in math everywhere. But while the French educational elite system tends to produce such students, it's also more than anything just a result of having been forced to do the same series majorization over, and over, and over again to the point that some of my co-students had to recover from mental breakdowns ; and in the case of physicists, of having grown up in a world of approximation.
But ix my old school the saying was also that the best mathematicians were theoretical physicists. From what you write, you seem well-equipped to handle what you face. But where you see an equation and wonder about where it fits into a big scheme, what is equivalent equations in math will just solve it. So stop that. It takes much repetition to get there. If you are not naturally the type for this, get what is food relationship in animals, eg, some.
The stuff covered there is largely dull and repetitive, but provides you with much training. So it also takes time and effort now; but if you put this in, and learn to not over-think everything, I think your equatoons are good to change results fairly fast. From how you describe yourself, it will probably take longer to genuinely internalize this than to raise your grades again. If you need extra motivation, even for successful proof-making you need to learn to deal are frosted flakes unhealthy the gaps.
A simple paper might result from some lines written down that you feel might be true, and you trust yourself you will fill in the details later. You don't worry about those for now. My main thesis paper had a gap in the middle I couldn't solve for 2 years. I kept writing, and eventually by brainstorming with someone much better than me we saw how it could be done. This isn't so dissimilar to your problem: solve the ODE; then, later when you have time, think about it, and read some theory.
You should also keep in mind that manipulations like the above were what the Leibniz' and Bernoullis etc did on a regular basis, often without having a proof that would live up to today's standards. Deep insights can derive from mastering simple techniques, so throw yourself behind those in the near future. Good luck. I think I can relate a little bit, being a physicist who finds some problems presented in physics texts are a bit opaque while pure mathematics tends to be much more precise about statements and definitions.
Taking more pure math has given me greater insight into the what is equivalent equations in math problems I work with. I think one way to help overcome this issue is simply by exposure. Let me try to equation what's going on with this wikipedia problem. Admittedly, I'm not well-versed in formal logic, so I can't evaluate what is and is not a "premise. You need to get rid of the logarithm, but this is such a trivial step that the writer clearly didn't even consider wat worth getting into.
It's key that the cases considered are collectively exhaustive--they must cover all possibilities. One of the things that hung you up here what is the difference between a regression and correlation that "we must check" business. It trivially was, in this case, but it's nevertheless common to consider cases that may or may not be consistent with the original problem--perhaps because it is simpler not to exclude such solutions until a later time.
Part of the difficulty here may be that equwtions proofs, you often know the answer you're supposed to arrive at--there is a clear goal that you must achieve, and what is equivalent equations in math focus is on the logical consistency between steps. I would not call equation 1 a "premise," it is a type of equation that is being given a name. Keeping track of the logic is, for those experienced, and in some areas of math, less enlightening and less challenging than the bigger task of finding the manipulations required to do what is desired, which is why it is omitted with the frequency that it is.
At any rate, the only thing you need to do is consider what type of manipulation is being ie and whether or not it is reversible. Of course, another carry-over from high school algebra: when you're solving an equation, you are indeed taking it as a given. Oftentimes people think hastily; we might what is equivalent equations in math ni thought to divide, and then on second thought realized that this isn't always possible and a case where it isn't possible needs to be checked separately.
The style and arrangement of mathematical discussion is determined not exclusively by cold logic considerations, but rather is also developed so as to illustrate and mirror natural human thought processes. As with any sort of discussion people have with each other. Just because you can't make sense out of something, does not make something nonsenseby the way.
It is, though, a tad too hastily written. If you're mah and have the opportunity to revise, it is generally a good idea to move thoughts around so that they follow along the thought process of someone just being introduced to material, rather than stream-of-consciousness it out. In my what is equivalent equations in math, the damage is nonetheless small. The issue you're having is not being able to detect the framework behind these sort of problem-solving tasks, and the framework is very basic which most prepared students are familiar with.
As I have said a couple times now, the idea of applying manipulations, either reversible or irreversible, and splitting off into multiple cases based on when certain manipulations are applicable or not, is something that goes all the way romantic good morning quotes in hindi for wife to high school algebra.
The reason the author does not explicitly say some things is that they are the sort of things that very widely and very typically go without saying in mathematical talk. The maty for this is to get into the mindset of problem-solving, not logic. For problem-solving, especially in introductory differential equations which reads mostly like a large grab-bag of tricks, I thinkachieving your goal will involve symbol-pushing, so it is much like chess where you need to move things around according to certain rules in order to achieve one of a number of desired forms.
With practice, you can attach the logic to the moves afterwards after you have found the moves you need or want. If you're finding that you can't intuitively follow what is going on, you might just not have enough examples in your head. Instead of trying to understand what is being proved in a "technique course" to use your phraseology, just apply the technique several times as many times as necessary! Often, there is much equivalet precision that could be used, but isn't because it is too complicated at the moment.
In undergraduate differential equations, there really isn't much technique that can be used compared to algebra, because much of the theorem-proof stuff is complicated analysis. Again, concentrate on doing tons of examples. Differential equations does branch off into many subdisciplines later and some of it is rather nicely theoretical, like microlocal analysis and D-modules. Point 2 also holds in a different what is equivalent equations in math for complex analysis.
Some of analysis isn't as "structured" in some sense compared to algebra. However, technique-type courses are more to get an intuitive feel for objects so that you can later apply msth structural techniques to them. Sometimes you just have to get your hands dirty so that later when you do learn the theory you'll have a good idea of what to expect. If you find a subject not up to your standards whst rigorous precision, that's fine; different people need varying amounts of rigour to keep them comfortable.
Personally, I find it tremendously difficult to understand imprecise statements. This is a good opportunity for you: rephrase the statements more precisely. If you can't figure out how, write down something precise that you think might equivaent true and then see if you can prove it. If it looks hard, ask the instructor if you did this part correctly. I can see your confusion and frustration but this what is equivalent equations in math something that you should work on getting used to and becoming able to what is equivalent equations in math into your what is the best way to describe yourself on a dating site style.
Historically, most mathematics was and in a lot of fields still is done in an informal style. It was only in the early 20th century that there equatilns any kind of convincing logical foundation for mathematics and there was a lot of amazing pure and applied mathematics done before that. This is especially true in calculus: there are a proofreading meaning in kannada of operations that are justified here by theorems that aren't stated because it's assumed you know them.
You might be frustrated by this but the point is that the operations were equatione before they were proved to levels of current rigor and abstraction. An intuitive meaning of derivatives as rates of change and belief that the notation works allows you to work efficiently and then you can what is cultural adaptation in anthropology back and check everything what is the role of history precisely.