Applied Mathematics vs. Science

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What are the differences between a branch of Applied Mathematics that deals with a particular branch of science and the actual branch of science itself?

For example: Fluid Mechanics is a branch of Physics which a lot of Applied Mathematicians are doing research in. What exactly is the difference between the Applied Mathematician's approach to Fluid Mechanics and the Physicist's approach to Fluid Mechanics?

Or, more appropriate for my own studies: The differences between a Mathematical Biologist studying stochastic processes in Biology and a Biophysicist doing the same thing?
 

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  • #2
Choppy
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Short answer: their titles.
 
  • #3
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Yes, but couldn't you postulate that Physics is essentially Applied Mathematics?
 
  • #4
WarPhalange
No, because physics does experiments and applied math deals with much more than science.
 
  • #5
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I like the other responses from the other people... pay attention to those...

Let's consider the fluid mechanics example. My understanding is that the applied math people who work in fluid mechanics are distinct from the physicists/engineers because:

1. they have a concern/curiosity for the rigor of the model, for example, is the description also continuous and differentiable, (in addition to being experimentally accurate)?

2. the math people are much more restricted to theory and computational study, whereas a physicist or engineer would tend to be more laboratory proficient, or more closely tied to his laboratory colleagues

3. the applied math people maybe have a bit more skill with the equations, getting new results, manipulating and studying the equations and numerical analysis, but they probably have a bit less skill and emphasis on intuition and just plain "good-ol understanding". In other words, the applied mathematician maybe doesn't have as good of a feel for the subject; they treat it a bit more abstractly.

That's all pretty general and by no means complete, but that's my sense for the answer at this point.
 
  • #6
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My understanding was that applied mathematicians are more interested in creating existence and uniqueness theorems for the models, while physicists are more interested in specific solutions and their physical interpretation.
 
  • #7
i'm also seriously wondering whats the difference as well. i would like more input into this
 
  • #8
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Short answer: their titles.
Not true.

First of all, it's going to depend where you are. For example, Applied mathematics in Britain was of course influenced by Newton (early on), then the Cambridge people in the 19-20th century, and then afterwards, Manchester in the 1950s. This had the effect of keeping it very much applied. In addition, applied maths is very much linked to industrial maths in Britain (with these Industry Study Groups, first initiated by Oxford in the 20th century).

In contrast, applied maths in America (and by extension, Canada) was more influenced by immigration of Germans and Russians. This had the effect of making it more pure. While some mathematics departments in the US have applied groups, they would perhaps be lumped into the Pure groups in the UK. So an applied mathematician coming over from UK to the US might even consider working with engineering or physics departments.

Then of course, you have variation between institutions. Doing applied maths at MIT or Brown is not the same as doing applied maths at say, Berkeley or Yale.

In my view, the simplest way to differentiate the physicist and engineer and the applied mathematician is by the following example.

Suppose you want to study the path of a ball thrown through the air. Both the physicist and the mathematician would know that it's modeled by a quadratic. Both would seek solutions -- numerical, analytical, or otherwise.

At this point, the mathematician may ask -- "Gee, instead of a quadratic, what happens if it's a cubic?"

To me, this fork in the road is the easiest way to distinguish between the two. Both fields study problems that are physically motivated. However, in mathematics, you tend to be more interested in the mathematical aspect of things: methodologies, structure, and so on. Your research will proceed according to what you consider mathematically interesting. Physical relevance -- while a bonus -- is not always demanded.

For example, I work on free-surface flows. Some of the regimes I'm interested in (low-speed, for example) are not priorities in Engineering applications. However, the mathematical structure is what I'm interested in.

Perhaps another way to think about it is through the following story: In the 1970s, the National Research Council of Canada realized they needed more mathematicians on their staff. So they decided to hire them. Today, there are almost no mathematicians working at the NRC.

Why? These people were required to justify -- on a scheduled basis -- the scope and expectations of their work (for funding purposes). But they could never do it! As an applied mathematician, you're often not confined to this strict adherence to physically related phenomenon...plus you're not always sure of where you're going and when you're going to be there.

My understanding was that applied mathematicians are more interested in creating existence and uniqueness theorems for the models, while physicists are more interested in specific solutions and their physical interpretation.
I'm an applied mathematician. I (and almost everybody I work with) have very little interest in existence/uniqueness theorems. I have never actually done a proof in the context of my research. In fact, of the literature that I read, I would say 98% (sort of random figure) have no theorem/proofs.

I can't actually name any theorem/proof paper I've read in the last 2-3 years. Okay, that's a lie. I can name one. Written by a Russian.
 
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  • #9
I'm an applied mathematician. I (and almost everybody I work with) have very little interest in existence/uniqueness theorems. I have never actually done a proof in the context of my research. In fact, of the literature that I read, I would say 98% (sort of random figure) have no theorem/proofs.

I can't actually name any theorem/proof paper I've read in the last 2-3 years. Okay, that's a lie. I can name one. Written by a Russian.
nice, i want to avoid rigorous thoremes/proof as much as possible
 
  • #10
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I'm not sure I agree with rsq_a. Choppy seems to hit it on the spot though.

Everything is connected, we are the one's who fracture it into separate subject areas and isolate ourselves to these areas. The questions are the same from whichever area you study it from and the answers will be based on your history and the education you received not based on the faculty you currently reside in.
 
  • #11
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At this point, the mathematician may ask -- "Gee, instead of a quadratic, what happens if it's a cubic?"
P.S. I'd be a shame if a physicist did not ask this same question, and in practice, he would.
 
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