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Physical motivation for integrals over scalar field?

  1. Feb 5, 2016 #1
    I'm looking for good examples of physical motivation for integrals over scalar field.

    Here is an example I've found:

    If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral

    It appears to me that the final temperature of our object would depend not only on its path (i.e. the image of the curve) but on the speed as well (if it spends a lot of time in an area with low temperature, it won't be reflected in its path but it clearly would be reflected in its final temperature). So it looks like the value of the integral would be parametrization-dependend (but it shouldn't).

    So I have two questions:
    1. Am I right that the temperature example is off?

    2. What are some good examples of physical motivation for integrals over scalar field? (If possible, don't assume any knowledge of physics.)
    Thank you in advance!
  2. jcsd
  3. Feb 6, 2016 #2


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    Impossible to say unless you specify the line integral. What makes you think the line integral cannot be made velocity dependent? You could very well also give the velocity as a function of the position in order to solve this. Although in reality it is not going to be a simple integral, the heat transfer generally depends also on the object's temperature.

    So you want physics examples without physics? That sounds like a contradiction in terms.

    This kind of integrals are probably the most intuitive ones you can find. Integrate a density and you get a total amount. For example, integrate mass density and you get total mass, integrate energy density and you get total energy.
  4. Feb 6, 2016 #3


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    I don't know whether this helps. However, I've asked myself how the average temperature on the earth's surface is calculated? There often is a number around - I think 15°C something - and how it changed through times. The integral was the only solution I could think of.
    I did not post it here, since probably someone would have shown up and started to debate the data stock, the timestamps, the distribution of measurements and so on and so on which won't let me expect a meaningful answer. Nevertheless, this global number is in this world and someone did the calculation despite all miseries.
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