Home | 18.022 | Chapter 11 | Section 11.1

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In writing down iterated integrals it is easy to confuse which variable is to be integrated first and which limits go with which variables.
There are two contradictory standard conventions for writing iterated integrals.

1. The inside out convention: The innermost integral sign and differential are to be done first, followed by the next innermost etc, as in the last example above.

2. The keep-the-differential-with-the-integral-sign convention: in which the variable associated with an integral sign and its limits is the one whose differential is closest to it.

Different authors use different conventions. Beware. And make sure you do what you do consistently. We try to use the inside out convention always.
Multiple integrals can be scalars, and will be so if f is a scalar function, or vectors if f is a vector (or even more obscure things at times).