# Examples of physical motivation for integrals over scalar field?

I'm looking for good examples of physical motivation for integrals over scalar field.

Here is an example I've found (source):

A rescue team follows a path in a danger area where for each position the degree of radiation is defined. Compute the total amount of radiation gathered by the rescue team along the path.

Does this example make sense? To me, it sounds like the total amount of radiation would depend not only on their path (i.e. the image of the curve) but on the speed as well, 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 radiation 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.)

• Think about things that are vector fields: examples of the top of my head range from calculating work (the full calculation of work requires a line integral) and calculating water flow (think of the liquids velocities as vectors) Commented Feb 6, 2016 at 3:29
• Also, see this image Commented Feb 6, 2016 at 3:31
• @BrevanEllefsen: I reckon vector fields are a bit different from physical point of view. So I'm asking specially about scalar fields. I've seen the image but it rather illustrates the geometric side of the issue (and not physical).
– Leo
Commented Feb 6, 2016 at 4:10
• Alright, then a key example is temperature, which is always a scalar field (temperature has no "direction"). 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 Commented Feb 6, 2016 at 4:25
• @BrevanEllefsen: To that I have the same question as to the radiation example: 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 on area with low temperature, it won't be reflected in your path but it clearly would be reflected in your final temperature). So it looks like the value of the integral would be parametrization-dependend (but it shouldn't).
– Leo
Commented Feb 6, 2016 at 4:36

Assume you have a point mass moving on a curve $\gamma$ now define on the curve the curvilinear abscissa $s$. You know that is given a velocity field on the curve. How long is the distance $d$ covered by the body in the time interval $[t_1,t_2]$?
$$d=\int_{t_1}^{t_2}v(s)\,\mathrm{d}s$$