Tricks To Solve Electrostatics Problems Pdf
Regenia Junke
I am confused about which is the method to solve electrostatics problems, if 1 or 2 (and, if both are correct, why and which is the best criterion for choosing it). I do not see the link between them.
I was just reading Chapter 6 of the Feynman Lectures on Physics Vol. II, which covers exactly this broad question about approaching electrostatics quite succinctly. You may be interested to read it too, it is available online.
I'm a sophomore in university and seriously feel that I'm bad at solving mathematical and algorithmic problems (be it discrete math, calculus or just puzzles). I noticed that I'm only good at solving questions that are similar to the ones that have been taught to us.
If the answer is no to all the above then I sort of blank out. I stare at it and force my brain to run through a wide variety of stuff, almost like a brute force attempt of solving it. Obviously that leads me to nowhere everytime. I simply can't think "outside the box."
I belonged to a school education system where we were made to do lots of different problems, but we were never told to try and understand the underlying theory behind the problems. This made me scared of math. What I basically had was a cookbook of a variety of wonderful recipes without realizing why I needed to add salt or sugar to a dish. May be you are facing the same problem? May be you are learning all these different techniques to solve problems without really understanding the theory behind why the problems can be solved using those techniques? Hence, because you don't understand the theory behind the techniques, once you get a problem that cannot be solved using the techniques you are familiar with, you get stuck.
While I agree with glebovg that trying to develop an intuition for how to write proofs is essential, I feel that you should make the effort to start reading proofs first. For instance, a book that really helped me understand Calculus was Spivak's Calculus. Try going through the proofs there, and learn the underlying theory. This is coming from someone who was in your position not too long ago.
Also, the issue of memorization is kind of a slippery slope. You will find that often even when you are trying to understand the theory, you will just have to memorize some computational techniques here and there. I think Terry Tao has a good post where he addresses the issue of memorization. I agree with him that certain basic things have to be memorized. For instance, you will have to memorize what the axioms of a group or a field are. I think memorization and understanding go hand in hand. Certainly your goal should not be to only memorize techniques to solve problems.
I think proving theorems really develops your thinking. Try to prove a few important theorems from calculus as well as discrete math, or try to understand someone's proof. Of course, the more you know the better, so that is why we say math is not a spectator sport. You need to do more than just the homework if you want to improve. Sometimes many results that you learn in, say discrete math, might seem confusing, but once you see why they are important in a different context, for example in number theory or algebra, you should remember them. To be honest, I think understanding and being able to prove theorems is actually relevant to math, whereas puzzles are just for fun. The best advice I can give is: Do not try to memorize math and simply remember everything for an exam because that way you might get a good grade, but you will forget everything a few days after the exam, instead try to understand why something is true. This way you will remember something practically forever, because you will be able to derive it when you forget.
I don't know about puzzles, so I write only about solving mathematics problems.In my experiences in this site, I find it far easier to solve problems in a field(like abstract algebra) I know well than in a field(like analysis) I know less.I think it's like walking in a town.If the town is where you live, you know every corner and you think you can almost walk with blindfold. On the other hand, if you are new in the town, you lose your way easily.
So the question is how we know a field well.Read textbooks, understand proofs, try to prove a theorem before reading the proof of a textbook. Reconstruct a proof without seeing a textbook.do exercises, try to find examples and counterexamples, try to find problems by yourself and solve them, etc.
I got a PhD and a postdoc in pure mathematics and I just can talk from my experience. I think that mathematical thinking can be improved with your experience solving problems and reading. For me, there are 2 options.
Option 1: improve your mathematical thinking by yourself. This means trying to approach the problem from all the possible points of view that you can imagine. Organize them, try to apply them one by one and draw a lot. Try to improvise and start solving similar problems in simpler versions. This is very hard to do, especially when you learn in a systematic way because this requires creativity but if you spend time doing this even if you don't solve it you can grow a lot and develop intuition.Warning! Do not spend more than 1 week with the same problem. Not all people solve problems quickly and that is just fine. If you cannot solve a problem after your hard work is a good idea to ask for help (books, mentors, the internet, etc) or just leave it in a special list and move forward. You will be able to solve that list in the future. Reading some comments reminds me that teaching to others the exercises that you can solve is a very powerful way to improve your mathematical thinking, it helps to organize your knowledge and discipline your mind. Please be patient with others. Some day someone will be patient with you and you will need it!!!
Option 2: improve your mathematical thinking using help. Some people may think that asking for help from others or books destroys your creativity and limit your mathematical thinking to the creativity of others. But only a few gifted can afford that. I think that there is nothing wrong with gathering some strategies of others to enrich your own bunch of tools. Consult a friend, professor, books or forums like this. When you ask for help the method to solve a problem is something that you haven't thought of, however, the new experience can help you to solve new problems in the future. When you are facing a new kind of problem and you do not have a clue where to start, look for examples and solved exercises. If you are in high school most likely there is a lot of reading material, examples and solved exercises for the topics that you are interested in.
I remember when I first got to college and was studying mechanical engineering. My high school education taught me the plug-n-chug method of thinking, so topics like differential equations, physics, let alone, linear algebra, dynamics, thermo, mechanics, etc. were really really difficult for me.
Somehow I struggled through it though, and graduated, but I always felt uneasy about having as solid of problem-solving skills in my educational foundation as I wanted on it. Especially since I was now working (tho my day-to-day work didn't require those specific skills). I ended up making a hack solution and practiced one math or physics problem a day on my own. I felt like I really came to understand those things since now I took the time to go through them myself, and see where all the formulas were derived from. Knowing that, I knew better when I could apply an equation, and in what manner.
I actually came across this site later: www.learnerds.com which pretty much was what I was looking for. An interesting (semi-realistic) math/engineering/science question a day with a good solution, and the authors are great at responding back to your comments, regardless of your level.
Over the last year or so of undergrad and first semester of grad school, I've completely atrophied my problem solving skill. At some point I became more comfortable with looking up a solution than trying to solve it myself. At this point my first instinct is to google something instead of trying to solve something. I need to fix this; it's already been affecting my performance and well being across the board. I'm almost instinctually aversive to trying to solve a problem by myself at this point. I feel like I've lost the ability to actually do math. I initially justified it by saying that my interest in math stems from my interest in the theory, and that I'm not particularly interested in problem solving. It's clear now that that was just cognitive dissonance. I need to fix this.
But I feel like it's not so simple either. I'm doing graduate level math after all. I managed to get into a fairly top level, rigorous program. I have performed well enough in the past that I managed to place ahead of my peers, and am doing relatively advanced courses (after all, I wouldn't have resorted to looking up stuff if it wasn't working well for me, until recently). As such, it seems like I already need a solid, strong problem solving capability in order to deal with my classes, which are quite demanding. So when I'm faced with HW or other problems, I'm unable to solve most of the problems even if I try really hard, because my problem solving skill is just so bad at this point, and I have to resort to looking things up once more. This further worsens my skill and on and on. It's a negative feedback loop. And I'm struggling to break out of it. I wanted advice on how to escape this loop especially. The idea of simply not looking stuff up is sound, but it's hard to follow through when I have only a finite amount of time before I have to stop thinking and submit my answers, or when I simply don't possess the capability anymore to try and solve the problem.
These days the idea of solving a graduate level HW set seems impossible to me, and I'm just incredibly lucky graduate level courses tend not to have exams. It's reached the point where it's threatening my future in my PhD program so I really do need to fix it. Googling my way through life isn't possible (or desirable either). I really am desperate now.I feel like a lost cause at this point, like the damage has already been done to me, and I can't really fix it without going back to undergrad or something.
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