Usages of calculus.
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 The Milkman
 Posts: 129
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Usages of calculus.
So I've managed to get a very good grasp on some very basic pieces of calculus. Mainly, the derivative, the integral and the fundamental theorems. I understand some important physical usages they have (i.e., the derivative at a given point on a function is the rate of change at that given point). The only thing I don't get is their usages theoretically. What can you achieve by taking the derivative or integral of a function? What does it show?
I'm looking for a more broad idea. I know the whole l'Hopital's rule as a theoretical use of the derivative, as an example.
Thanks in advance.
I'm looking for a more broad idea. I know the whole l'Hopital's rule as a theoretical use of the derivative, as an example.
Thanks in advance.
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Re: Usages of calculus.
Calc has vast importance for the physical sciences. For example, in physics, velocity is the derivative of position and acceleration is the derivative of velocity. Also http://en.wikipedia.org/wiki/Calculus#Applications
For theoretical math? I'm not sure, but I'm sure people far smarter than I on this forum will be.
For theoretical math? I'm not sure, but I'm sure people far smarter than I on this forum will be.
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Re: Usages of calculus.
It isn't an overstatement to say that all of physics and engineering is an application of calculus. Plus a healthy percentage of the remaining sciences, economics, finance, and statistics. As far as the theoretical sciences, essentially anything that is developing a model that would eventually be applied to one of those fields is pretty certain to be tied to analysis.
It is as you say. The derivative of a function at a point is the instantaneous rate of change. And the direct integral of a function over a range is the accumulation of "stuff" over a period of time where we are measuring the instantaneous rate of accumulation. Those are two extremely relevant matters to be asking about.
It is as you say. The derivative of a function at a point is the instantaneous rate of change. And the direct integral of a function over a range is the accumulation of "stuff" over a period of time where we are measuring the instantaneous rate of accumulation. Those are two extremely relevant matters to be asking about.
 The Milkman
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Re: Usages of calculus.
Why is it important though?
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Re: Usages of calculus.
Well, if you don't think that all of physics and engineering is important, then we will kindly take all our bridges and airplanes back.
 The Milkman
 Posts: 129
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Re: Usages of calculus.
No, that's not what I meant.
Why is this relevant on a theoretical level? Is it at all?
It is as you say. The derivative of a function at a point is the instantaneous rate of change. And the direct integral of a function over a range is the accumulation of "stuff" over a period of time where we are measuring the instantaneous rate of accumulation. Those are two extremely relevant matters to be asking about.
Why is this relevant on a theoretical level? Is it at all?
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Re: Usages of calculus.
The Milkman wrote:No, that's not what I meant.It is as you say. The derivative of a function at a point is the instantaneous rate of change. And the direct integral of a function over a range is the accumulation of "stuff" over a period of time where we are measuring the instantaneous rate of accumulation. Those are two extremely relevant matters to be asking about.
Why is this relevant on a theoretical level? Is it at all?
In practice it quite often happens that it is easy to say something about the rate of change. For example, the rate at which something is cooled down or heated up by the environment is proportional to the temperature difference. This is an intuitively obvious fact. Using calculus you can then deduce that the temperature changes over time like a decaying exponential, which is not entirely obvious. There are so many processes in physics where it is easy to see the things that cause something to change, and this usually leads to a differential equation that you need to solve if you want to know the actual state at a future time.
Re: Usages of calculus.
I'm pretty sure that The Milkman agrees that calculus is of enormous importance in applied mathematics, as virtually all laws of physics can be expressed in the form of differential equations, but he wants to know how important calculus is in pure mathematics.
The best answer I can give is that calculus is important because we can do it. It allows us to describe interesting behaviours of a vast range of functions, primarily those functions that have wellbehaved derivatives. But I guess that's a bit like saying vision is important because it lets us perceive stuff that's not invisible. Luckily, we live in a world with lots of interesting visible stuff.
Even though pure maths can investigate any mathematical structures it feels like, historically there is a synergistic relationship between pure & applied maths, so it makes sense that pure maths investigates the foundations of the maths that the applied guys find useful. For more than 3 centuries, calculus has been a major force in describing the world mathematically, so it is important in both pure & applied maths. People used calculus to do all sorts of interesting things, but for most of that time the theoretical basis was rather dodgy, and it wasn't until the end of the 19th century that calculus was put on a firm theoretical basis, using the modern definition of limits. And building this foundation has lead to the development of various interesting topics in pure maths, including modern notions of infinity, and understanding functions that do not have nice derivatives.
The best answer I can give is that calculus is important because we can do it. It allows us to describe interesting behaviours of a vast range of functions, primarily those functions that have wellbehaved derivatives. But I guess that's a bit like saying vision is important because it lets us perceive stuff that's not invisible. Luckily, we live in a world with lots of interesting visible stuff.
Even though pure maths can investigate any mathematical structures it feels like, historically there is a synergistic relationship between pure & applied maths, so it makes sense that pure maths investigates the foundations of the maths that the applied guys find useful. For more than 3 centuries, calculus has been a major force in describing the world mathematically, so it is important in both pure & applied maths. People used calculus to do all sorts of interesting things, but for most of that time the theoretical basis was rather dodgy, and it wasn't until the end of the 19th century that calculus was put on a firm theoretical basis, using the modern definition of limits. And building this foundation has lead to the development of various interesting topics in pure maths, including modern notions of infinity, and understanding functions that do not have nice derivatives.
 intimidat0r
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 Joined: Thu Aug 02, 2007 6:32 am UTC
Re: Usages of calculus.
There are many theoretical applications of derivatives and integrals.
Proof that pi is irrational: http://www.lrz.de/~hr/numb/piirr.html
The proof that e is irrational (the one I know anyway) relies on Taylor series, which relies on derivatives and integrals. The proofs I know of pi's and e's transcendentalness (?) also heavily use concepts learned in calculus. Also Taylor series and related concepts can be used to estimate many functions to an arbitrary precision.
Defining log, exp, sin, cos, etc. is basically impossible without integrals and derivatives.
Also yes, basically all of physics needs calculus (or math at a higher level than calculus) very very much.
Proof that pi is irrational: http://www.lrz.de/~hr/numb/piirr.html
The proof that e is irrational (the one I know anyway) relies on Taylor series, which relies on derivatives and integrals. The proofs I know of pi's and e's transcendentalness (?) also heavily use concepts learned in calculus. Also Taylor series and related concepts can be used to estimate many functions to an arbitrary precision.
Defining log, exp, sin, cos, etc. is basically impossible without integrals and derivatives.
Also yes, basically all of physics needs calculus (or math at a higher level than calculus) very very much.
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Re: Usages of calculus.
Here's a simple example. The local minimums and maximums of a function are the zeroes of that function's derivative. Try it out for yourself with a polynomial.
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Re: Usages of calculus.
If I understand you correctly, your question is, in some sense, "Do many mathematicians use calculus to prove things, or is it mostly used by engineers to calculate things?"
Well, using knowledge of derivatives and integrals, you can divide functions into various classes (for instance, L^{2}space is the space of functions f such that [imath]\int_{\infty}^{\infty}f(x)^{2}dx[/imath] is finite, and the Schwartz space is the space of functions that decrease really freaking quickly), and then you can use the properties that define that class to make some rather general statements. For instance, you can show that all Cauchy sequences in L^{2}space have limits in L^{2}space. ...and using knowledge of the Schwartz space, you can define distributions and transform a lot of calculus problems into algebra problems!
...and as others have mentioned, series such as the Taylor series have a wide range of uses, as do many transforms which involve integrating functions over an infinite range.
Well, using knowledge of derivatives and integrals, you can divide functions into various classes (for instance, L^{2}space is the space of functions f such that [imath]\int_{\infty}^{\infty}f(x)^{2}dx[/imath] is finite, and the Schwartz space is the space of functions that decrease really freaking quickly), and then you can use the properties that define that class to make some rather general statements. For instance, you can show that all Cauchy sequences in L^{2}space have limits in L^{2}space. ...and using knowledge of the Schwartz space, you can define distributions and transform a lot of calculus problems into algebra problems!
...and as others have mentioned, series such as the Taylor series have a wide range of uses, as do many transforms which involve integrating functions over an infinite range.
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Re: Usages of calculus.
Anything in math with the word "analysis" relies on some calc.
Also, let's say you decide you don't like all this continuous function business. Leave that to the engineers, you can go be a number theorist. Stick with the integers. Nice, solid, well behaved. It will not be long till you hit things like the all important
http://en.wikipedia.org/wiki/Prime_number_theorem
the quite classical
http://en.wikipedia.org/wiki/Gauss_circle_problem
the quite recent and impressive
http://en.wikipedia.org/wiki/TaoGreen_theorem
Similarly, if you decide you just care about Euclidean geometry, it will not be very long until you start asking questions where the answer requires analysis. And then you may be tempted by the delights of differential geometry...
Also, let's say you decide you don't like all this continuous function business. Leave that to the engineers, you can go be a number theorist. Stick with the integers. Nice, solid, well behaved. It will not be long till you hit things like the all important
http://en.wikipedia.org/wiki/Prime_number_theorem
the quite classical
http://en.wikipedia.org/wiki/Gauss_circle_problem
the quite recent and impressive
http://en.wikipedia.org/wiki/TaoGreen_theorem
Similarly, if you decide you just care about Euclidean geometry, it will not be very long until you start asking questions where the answer requires analysis. And then you may be tempted by the delights of differential geometry...
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 The Milkman
 Posts: 129
 Joined: Fri Jul 10, 2009 12:19 pm UTC
Re: Usages of calculus.
That's rather interesting because I AM interested in number theory. I knew about the prime number theory, but didn't realize that calculus was involved in it.
I think I'm starting to get the idea now...
I think I'm starting to get the idea now...
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Re: Usages of calculus.
One neat use of calculus that I don't think has been expressed yet is that integrals can be used to find bounds on sums. In particular, if you notice that a certain sum you're looking at happens to be a lower riemann sum for some function, then you can bound your sum by the integral of the function. This doesn't sound like it's that important, but it's a really useful trick.
Re: Usages of calculus.
An oddball use of differentiation is as such: If you differentiate a function and find it to always equal zero, you know that the function is a constant function, which can be used i.e. as a method to prove identities.
Re: Usages of calculus.
Patashu wrote:An oddball use of differentiation is as such: If you differentiate a function and find it to always equal zero, you know that the function is a constant function, which can be used i.e. as a method to prove identities.
Only on every connected component of its domain. For instance, the Devil's Staircase is continuous on [0, 1], and has zero derivative almost everywhere (it is differentiable everywhere except the Cantor set), but is nevertheless far from constant.
Re: Usages of calculus.
The Milkman wrote:That's rather interesting because I AM interested in number theory. I knew about the prime number theory, but didn't realize that calculus was involved in it.
I think I'm starting to get the idea now...
Pick up Prime Obsession, it's all about analytic number theory (number theory + calculus).
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