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Calculus and Technology

  1. Aug 29, 2009 #1
    OK. Weird question.

    What is a technology that couldn't exist without calculus?

    Like, a specific engineered artifact -- a bridge, a segway, a waterjet cutter -- that couldn't have come into being without calculus.

    Not necessarily a hard question, just something I'm pondering for fun. Trying to get a better grasp on the real value of things like mathematics. (One thought I'm having is that if you were intuitive enough, you could engineer anything without explicit math/science. Say, if you started building small bridges as a kid, and by the time you were an adult you had gotten a good "feel" for it, and could do it without any book learning. So I'm thinking about whether math is truly necessary.)

    But... that's kinda off the deep end :D. Let's keep it as "What is a technology that couldn't exist without calculus?"
     
    Last edited: Aug 29, 2009
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  3. Aug 29, 2009 #2
    Freeman Dyson on the role of failure:

    "You can't possibly get a good technology going without an enormous number of failures. It's a universal rule. If you look at bicycles, there were thousands of weird models built and tried before they found the one that really worked. You could never design a bicycle theoretically. Even now, after we've been building them for 100 years, it's very difficult to understand just why a bicycle works - it's even difficult to formulate it as a mathematical problem. But just by trial and error, we found out how to do it, and the error was essential."

    I got this from wikipedia (http://en.wikipedia.org/wiki/Freeman_Dyson) and the original quote is supposedly from:
    http://www.wired.com/wired/archive/6.02/dyson.html
     
  4. Aug 29, 2009 #3

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    Have you ever seen bridges built by pre-scientific societies? Sure, you could probably build something that can be called a 'bridge' just by trying out random things and seeing what works best, but no matter how long you do that, you will not be able to get even close to something like a modern bridge. All modern engineering structures are based on scientific principles (expressed in the language of mathematics). In fact, if what you're interested in is the question of whether math is truly necessary as you say, then a bridge is a bad example. Can you build a computer without a knowledge of science? A television? A Plane? A Satellite?
     
  5. Aug 29, 2009 #4
    He didn't say "science" he just said "calculus". A lot of science (experimantal method, etc.) doesn't need calculus.

    Plotting trajectories for spacecraft ... differential equations are used.

    Perhaps semiconductors (transistors and such devices used within electronics) would not have been invented without predictions from differential equations describing quantum mechanics.
     
  6. Aug 29, 2009 #5
    Is this the thing about a rocket's mass changing as it burns fuel, changing its fuel usage?

    Although, just to be difficult: what if you made a computer simulation of that process to determine the trajectory? Could you do it without calculus then?

    Huh. I know next to nothing about QM, but it's interesting it's needed (or, might be needed) for transistors.

    What I'd really like is to find a handy little example of some specific device that needs calculus...
     
  7. Aug 29, 2009 #6
    The thing is you would be using concepts of calculus within your computer simulation. If you weren't using calculus, then there is no way you can be certain with your simulation.

    Antennas, Transmission Lines (The Telegraphist Equations), Modern Communication Systems, Etc. There are many more examples, but those are just a few that I'm familiar with.
     
  8. Aug 29, 2009 #7
    Hmm... I suppose it depends on what you mean by calculus.

    Let's say we're trying to figure out the rate the fuel is burning, and that it depends on the mass of the craft.

    Now, if you knew an equation to relate fuel usage, speed, and mass, you could just have your program change the MASS and FUEL_LEFT variables every second the program is running.

    I would say that isn't using calculus. We don't have a concept of "limit" or "derivatives", some way to get an equation "a priori" for the craft's fuel usage rate. Only a simulation, done over small intervals, using an equation (aka, algebra).
     
  9. Aug 29, 2009 #8
    The methods of calculus are used to get a new equation from knowledge of another equation. In particular, it is mostly used in physics to get equations for dynamics from empirical equations of static phenomena (differentiation) or to get an equation for a static entity from knowledge of how a system evolves (integration; solving differential equations). Once you have these equations, you can program them into your computer, or apply them directly. The calculus simply generates the equations; calculus is not necessary to actually carry them out.
    However, knowledge of calculus means you need only know one equation and you immediately have access to an entire family of related equations. It also allows you to approximate complicated relationships using numerical analysis.
     
  10. Aug 29, 2009 #9
    OK, I see. Wikipedia says geometry is the study of shape, algebra the study of equations, and calculus the study of change. From that definition, I can see how even the program/simulation would be the calculus.

    I guess it seemed odd to me (and still does), to call it calculus when you're not using equations. What the program is doing isn't a complex concept: it's just computing differences across small intervals. Then again, I guess the idea isn't said to be complex. It's just that it took awhile for anyone (Newton/Leibniz) to actually work out how to do it formally (e.g. differentiate a general function like x*sin(x), rather than working it out numerically at every point).
     
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