dextercioby said:
Mathematics is an ESSENTIAL tool for a theoretical physicst...
Daniel.
P.S.I would have quoted Feynman,a theorist,not Galileo,an experimentalist...
Mathematics is an essential for any physicist, no matter what field you're in.
As the grandfather of modern science, Galileo, like Newton, was both a brilliant theorist and an extremely skilled experimentalist.
ZapperZ said:
But using it ".. to describe nature and to make our results quantitative..." is precisely using it as a "tool". This description of mathematics, as used in physics, does NOT demean nor diminish its importance. Without it, physics has no language and thus, unable to express itself accurately (try describing Gauss's law in words!).
We use human language as a "tool" to communicate with we talk to each other. Most physicists use mathematics as a tool in their work. No one should be offended by this, least of all, mathematicians, considering that without mathematics, physics will be mute.
Fair point. It just didn't feel right to call something essential a 'tool'. Like your analogy; we cannot communicate without some sort of language. Likewise, we cannot do physics without mathematics, therefore it's an integral part of it.
But aw'right, an essential tool is fine with me.
If we don't do that, we end up NOT doing physics, but end up learning more mathematics than what most math majors would need. Students of physics do not have the time, the patience, nor the inclination to delve into mathematics that deeply - that is why we are not math majors. You are also forgetting that knowing what the "physics" is behind the mathematics allows for the simplification of the problem that isn't obvious from the mathematics.
There's nothing wrong with using physical arguments, quite the contrary. It's the derivation of many equations and such that could be done more carefully.
This could be just me, but whenever there's a step in a derivation I don't precisely understand as to 'why' it is allowed or simply don't quite get it, I get a very uneasy feeling about it. Some sense of incompleteness in my understanding.
Maybe I could conjure an example. Like the equation of motion for a rocket ejecting mass (fuel) out of its rear:
"Well, in an interval between t and t+dt the amount of fuel exhausted is |dm|=-dm (because the mass of the rocket decreases), while the mass of the rocket is m+dm and its velocity \vec v+d\vec v.
The momentum of the system at time t is:
\vec P(t)=m\vec v
and the momentum at time t+td is:
\vec P(t+dt)=\vec P_{\mbox{rocket}}(t+dt)+\vec P_{\mbox{fuel}}(t+dt)=(m+dm)(\vec v+d\vec v)+(-dm)(\vec v+\vec u)
\vec u is the velocity of the exhaust gases wrt the rocket.
The change in momentum in the time interval dt is:
d\vec P=\vec P(t+dt)-\vec P(t)=m d\vec v -\vec u dm
where we have dropped the second order term dmd\vec v.
Divide by dt to get the change in momentum, which equals the external force.
Rewrite to get:
m\frac{d\vec v}{dt}=\vec u \frac{dm}{dt}+\vec F
In the case of no external force (no gravity in outer space) \vec F=0. We can multiply both sides by dt/m and integrate to find:
\vec v=\vec v_0 +\vec u \ln\frac{m}{m_0}"
I`m not saying the result is wrong or questionable. It's very plausible if you physically interpret this answer.
I find the derivation quite horrid. Things are done I was told that weren't allowed, like treating dm/dt as a fraction. It would be much more elegant to set up a differential equation and solve it. This doesn't even have to be done in a physics class, but in a lecture on DE's.
I`m sure I can think of more examples, but something like this make me go:
-< (Is this kosher?)
