Our coverage of multiple integrals will actually parallel our earlier coverage of Riemann integrals pretty closely. Only now we have to change our notion of “interval” to a higher-dimensional version.
For the moment, the only shapes we’ll integrate over will be closed rectangular -dimensional parallelepipeds with sides parallel to the coordinate axes in . In one dimension, these are just coordinate intervals, but there aren’t many other obvious shapes to use in that case. In two dimensions, we are (for the moment) throwing out circles, ellipses, triangles, and anything else that’s not a rectangle; and even rectangles that are tilted with respect to the coordinate axes! Don’t worry, we’ll get them back later.
Anyway, such a shape in -dimensional space is the product of closed intervals:
In this context, we’ll write this as for and . This looks identical to, but is not to be confused with, the closed line segment from to we used in the mean value theorem. I’ll try to keep them separate by always referring to this rectangular parallelepiped as an “interval” and never using that term for a line segment. Of course, in one dimension the two are exactly the same.
Now we define a partition of an interval. We’ll do this just by partitioning each side of the interval. That is, we pick points
where we cut the th side into pieces, and each of the could be different. If we index the subintervals of the partition by through , we can write
picking out the th slice of the th side. This cuts the whole interval up into a bunch of smaller pieces, which are themselves rectangular parallelepipeds. Again, we have the notion of a tagged partition, which picks out a sample point for each subinterval. And again we say that one tagged partition is a “refinement” of another one if every partition point of the coarser partition is also one in the finer partition on the same side, and if every sample point in the coarser partition is one in the finer partition as well.
There’s a lot of numbers here and a lot of notation, but relax: most of it is going to go away very quickly. Go back and look at how partitions, tagged partitions, and refinements were defined in one variable, and just think of partitioning (and refining) the one-dimensional interval that makes up each side of the -dimensional interval. Of course, like I said before, the number of parts on each side might have nothing to do with each other. Further, there’s no reason for the tags to have anything to do with each other. One might think of tagging the partition on each side of the interval, and then letting the sample points in the subintervals have those tags as coordinates. This is a perfectly valid way to come up with a collection of sample points, but the sample points don’t have to arise in this manner at all, so long as there’s exactly one within each subinterval.
So, just like in the one-dimensional case, the collection of all tagged partitions of an interval form a directed set, where we say that if is a refinement of . And again we define nets on this directed set; given a function defined on the interval and a partition , we define the Riemann sum
by sampling the function at the specified point in the subinterval , multiplying by the -dimensional volume of the subinterval (which is just the product of its side-lengths), and summing this over all the subintervals in .
And again, as before, if this net converges to a limiting value , we say that is Riemann integrable on the interval and we write
in three different notations, depending on what we want to emphasize. The first emphasizes the function as an object in and of itself; the second, the function’s dependence on the vector variable ; and the third, the function’s dependence on each individual real variable. In two and three dimensions, we often write the “double integral” and “triple integral” as
as visual reminders that these are multiple integrals. This gets unwieldy very quickly, and so we usually just write one integral sign for each integration.