No specific questions here, *All* opinions welcome

[TheNoobie001]

Hello everyone, my name is Evan operating under 'thenoobie001'.[this is also my xbox gamertag if there are any CoD fans out there :)]. I have recently joined this site and I anticipate that I will be doing a lot of forum searching and reading here at the beginning of my membership since my knowledge of math and physics is at what most of you might call a "beginner" level. I will try my best to search the forum looking for answers before I create a new thread on a subject that has potentially been through the ringer, so to speak.

With that being said, on to the reason for this post in career guidance.

I am 30 years old and I am getting back to my roots of my love for math and physics. I graduated high school from the Marine Military Academy in 2003 where I learned math and physics through the books of John Saxon. I completed Saxon: Algebra 1, Algebra 2, Pre-Calculus, and Physics. Algebra 1, 2 and Pre-Calc incorporates geometry and trig into these three subjects over the years with a building block approach. The Physics book states in its preface, "This book contains 100 lessons and covers in considerable depth all of the topics normally presented in the first two semesters of an engineering physics course". So this is my background in math and physics, but keep in mind it was 12 years ago so I am a little rusty. I have purchased these books and I am now going back through them working all of the lessons. I am doing this in preparation for starting my BA in Mathematics(I am doing this online because I currently work offshore and this is the only way to earn and education without having to give up my current career, as long as I am gaining knowledge and moving forward I am perfectly happy with it.)

At some point I am going to have to take the plunge and transition into a traditional classroom style school when I am ready to work towards my physics degrees, but like I mentioned above, I can still make progress in math while earning decent money to save for my physics transition).

Now that you have a little bit of background on me, on to my plan. I personally feel that mathematics and physics are intertwined and that they should be studied together. That is my personal opinion and in no way an attack on anyone else's opinion. With this in mind I plan on earning undergrad degrees in both mathematics and physics, and then pursuing more advanced degrees in both. I understand this will take some time and I am perfectly alright with this. I am not pursuing these fields to simply earn a degree, but to gain knowledge and I fully expect to be learning until the end of my time :).

So that is basically the general direction I am going and I wanted to put it out there so there is a little background information on me. I welcome all opinions and advice, and if anyone has any more specific questions for me please feel free to ask.

Thank you all for your time. Have a great day.
TheNoobie001

 

[Greg Bernhardt]

Good luck, keep us updated!

 

[TheNoobie001]

Good luck, keep us updated!

Will do, thank you.

 

[TheNoobie001]

As I stated above I am going through my old mathematics and physics books in preparation to starting my Bachelors Mathematics program and I noticed something that... well for a lack of better words... bugs me.. The physics answers are more of an approximation while the mathematics answers are more exact. Then I did some looking around the forum and found this to be the norm rather than the exception.

I fully understand where this comes from; the significant digits and only being able to use answers as accurate as the instruments used to measure them. I am committed to both physics and math, but I prefer precision and accuracy as well.

Now to the reason of this post... Is there a branch of physics that caters to the more precise answers? Does either theoretical or experimental use more precise answers than the other? Also, I stumbled on a group known as "Mathematical Physicists" .. Can anyone give a more detailed overview of what a Mathematical Physicist does? Are they more theoretical in nature and work with manipulating physics equations of the various subfields of physics to help solve problems(which is what I think they do) or am I completely wrong on what I think? Also, I have noticed that mathematical physicists fall under "applied mathematics" which leads to another question... which route would be better suited at the undergrad level for mathematics, pure or applied? I know it falls under applied so naturally one could assume to pursue an applied route but if you are working to solve problems in physics through mathematical manipulation of equations it would seem that pure mathematics would be in order to bring more abstract thinking to the subject.

Just some thoughts and questions. I am not close to having to decide any route yet, but I would like to understand a little more about the routes before hand... if someone has the patience and time.

Thank you for taking the time to read this.
TheNoobie001

 

[axmls]

The physics answers are more of an approximation while the mathematics answers are more exact.

There's a reason for this.

The purpose of physics is to describe nature. We don't need accuracy to 50 decimal places to do that. I mean, we use Newton's laws to send men to the moon, whereas General Relativity is the more accurate theory. Why? Because wasting extra time solving Einstein's equations would mean a difference of about 1 cm.

That's the thing: when you demand more accuracy, you're usually demanding more computational time, and it really isn't worth it to know that acceleration of an object is 15.3728 m/s^2 as opposed to 15.373 m/s^2.

Another reason is that exact expressions don't always give more physical insight into what's going on. We could possible have an expression that will tell us the exact value of the electric field everywhere in a certain scenario, but we don't want to solve it. Instead, we make approximations (like assuming a wire is infinitely long) so that we can see the "meat" of what's going on without worrying about the fine details.

Or, as Feynman put it, the physicists deal with a few special cases and leave the number-crunching to engineers and applied mathematicians.

I believe mathematical physicists focus on new mathematical methods to solve problems in physics (though I'm not sure).

Every physical theory has limitations, and they're all inaccurate in some realm (even the Standard Model doesn't include gravity). We can write down equations that will give us as accurate values as we want. If we get accurate enough measurements for our constants or whatnot, we can plug those in and out comes an accurate answer. However, 1. we don't always need extreme levels of accuracy, 2. oftentimes equations themselves are products of approximations.

There are some cases where we can find closed-form solutions to Maxwell's equations, but physics deals with the real world, and in the real world, we always end up having to approximate.

 

[TheNoobie001]

There's a reason for this.

Thank you for your quick and detailed response. You have helped me understand why the two fields are this way. Again, thank you very much, it helps to put my mind at ease!

 

[axmls]

Thank you for your quick and detailed response. You have helped me understand why the two fields are this way. Again, thank you very much, it helps to put my mind at ease!

Quick example:

Sometimes, the equation you find for, say an electric field, is given by an infinite sum of sines and cosines. Sometimes, we don't know the exact value of the sum, but we don't need the exact value. Usually adding up a few of the terms and dropping off the remaining (miniscule) terms is quite accurate.

The more complex a system is, the harder it is to actually model it mathematically and find an answer that isn't approximate, because the real world's more complicated than an ideal massless-string and massless-pulley with no friction world.

In electrical engineering (which I study), we can represent square waves with an infinite sum of sines and cosines. If we use this as an input into a circuit, the output might also be such an infinite sum. We don't need the 5000th term of the sum--we just need an approximate value. For an electrical engineer designing a circuit, that's all we need.

Ultimately, here's a fun little example.

Let's say you have two charged particles sitting there lined up on the z axis with separation $d$, with the origin being in the middle of them: a positive charge $q$ and a negative charge $-q$. The electric potential (voltage) anywhere is given by
$$V(x,y,z) = \frac{1}{4\pi \epsilon_0} \left [ \frac{q}{\sqrt{[z - (d/2)]^2 + x^2 + y^2}} + \frac{-q}{\sqrt{[z + (d/2)]^2 + x^2 + y^2}} \right ].$$

That's what you get if you use the voltage formula. Now let's say the two particles are very close together--this is called a dipole. Well, that formula right there doesn't tell you very much about this case. It's still the same formula, but it doesn't give you any physical insight into how these dipoles behave. However, we can actually come up with an approximation if we assume $d$ is very small that gives
$$V(x,y,z) = \frac{1}{4\pi \epsilon_0} \frac{z}{r^3} q d$$
where $$r^2 = x^2 + y^2 + z^2$$
Now, that's much simpler, gives us some nice insight into how it acts, and it's pretty accurate whenever $d$ is small (as is the case with dipoles).

That's a case when approximations are useful--not only for calculation purposes, but also for physical insight (the approximated equation gives us insight into how dipoles in particular act). Now, I left some cool things out, but those are the kinds of approximations you'll be doing in physics (and they show up all the time).

 

[TheNoobie001]

I have not made it back that far in my physics book yet for these formulas but I have earmarked my book with a printout of your post so I can use it in my studies to help me. With that being said you have definitely illustrated a situation where the approximation is useful and helped me to understand why you would approximate something, regarding different scenarios.

Thank you again for your insight. Once I get to this section in my book I might have reply directly to this post to ask a question if one arises that pertains to this.

Thanks again for your help, you have helped me understand something and become more comfortable with the idea of approximating values. Kudos to you axmls.