So I was wondering if manipulating differential as fraction is fine.In Physics problems ,usually, what I see is if we have equations of magnetic field ,we consider a magnetic field caused by a small length dl and the magnetic field is dB and integrating over the whole length.it seems we are treating dB as a infintesimal quantity.However ,after browsing for a while,the answer is that the notation dy/dx is not a infintesimal quantity but rather an operation on the function y.I am confused as to how it is valid in physics to treat it as infintesimal quantities.(After learning it,my notion for physics just got crushed and am now confused).I have not learned multivariable calculus yet,but from the arguments I've read ,I found that interpreting them as fractions does not work anymore.But it seems it works fine when in 2 dimensions.In mechanics when we derive equation for rocket motion (ejecting mass) we interpret differential as small quantities (dm) and divide by small time dt.Sorry for the long story... I am confused right now  TL:DR : In physics ,dy/dx can be operated as if it was a small quantity divided by small quantity,though from what I've read it is not supposed to be a ratio.  When does interpreting them work and when does it not. Any help would be very much appreciated ...
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio.   I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe?
Dear Academic SE advisers,  I am a college sophomore in US with double majors in mathematics and microbiology. My algorithmic biology research got me passionate about the number theory and analysis, and I have been pursuing a mathematics major starting on this Spring semester. I have been independently self-studying the number theory textbooks written by Niven/Zuckerman/Montgomery, Apostol, and Ireland/Rosen on this semester. As this semester progressed, I discovered that I am more interested in the pure mathematics than applied aspects (computational biology, cryptography, etc.). I want to pursue a career as analytic number theorist and prove the Collatz conjecture and Erdos-Straus conjecture.   I have been thinking about doing the number-theory research on my university (research university; huge mathematics department). I have been self-studying the NT by myself and also regularly attending the professional and graduate seminars on number theory but I did not do any pure mathematics research as an undergraduate. Should I visit NT professors in my university and ask them about if I can do undergraduate research under them? If research is not possible (perhaps due to my lacking maturity), should I request of doing independent reading under them and later proceed with the research? How should I ask them? What should I address? If even independent reading is not desirable to them, what should I ask to them or do in my own?   As for my mathematical background, I have been taking Calculus II (computational) and discrete mathematics. I will be taking calculus III (vector calc.) on Summer, followed by Analysis I, Probability, Theoretical Linear Algebra on Fall 2015. As for my self-studying on this semester, I have been studying NT textbooks (mentioned above), proof methodologies, and basic linear algebra.   Thank you very much for your time, and I look forward to your advice!  Sincerely,  PK Did Sam's memories transfer to the displacee when he leaped? In the TV show Quantum Leap Sam Beckett would travel history and "right" wrongs, temporarily displacing someone native to that timeline in the process. Would the memories of what Sam did transfer to the person he displaced upon his leaping out, or would they wind up with amnesia for the time Sam was there? Using the code below I add text to end of file names which are:  1.mp3 2.mp3   and should be:  1text.mp3 2text.mp3   but it changed to:  1.mp3text 2.mp3text   Code:  @ECHO ON  FOR %%A IN ("C:\Users\user123\Desktop\mp3\*.mp2") DO (     CALL :RenameFiles "%%~A" "%%~NXA" ) GOTO EOF  :RenameFiles  SET fname=%~2 SET renname=%fname=% REN "%~1" "%renname%text%" GOTO E Print floating point values without leading zero How can mixed content compromise an entire HTTPS session? Google Chrome is probably one of the stricter browsers out there in terms of highlighting SSL errors.  Most browsers will point out when you are viewing an HTTPS page which loads insecure images/Javascript, I understand how this is a security issue - an attacker could have modified the non-SSL content in a way which gives them access to the secure content, injected some javascript for example.  However once you have viewed an HTTPS page with mixed content, Google Chrome will continue showing you the mixed content warning (a red line through the 'https://') for as long as you stay browsing the same domain, even for subsequent pages which only include other HTTPS resources. Only once you navigate to another domain, or open a fresh tab does it tell you you're secure again.  What is the danger it's protecting you from here? How could mixed content on a previous page actually affect the security of subsequent secured pages? 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As an example, if I'm in a SSH session to a server of mine and after X minutes of inactivity in the terminal, I want a command to start a desired program like cmatrix, and when I come back and press a key I will get back where I left. Behaviour of return statement in catch and finally I have a problem with verb structure In this sentence:     Four items are responsible for three-fourths of the overall spending around the world, the most prominent share being that of food.   Why after the comma, the verb "being" is used? Why it does not use "is"? What can I do to give the players the same feel their characters would have about wilderness travel? 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