42 Vector calculus
42.1 Surface integrals
Surface integrals are generalisations of the concept of a multiple integrals to integrals over surfaces. For example, consider a flat plate of area \(A\) heated to a certain temperature, and thus emits a certain amount of energy, \(E\), per unit area (per unit time). The total energy emitted per unit time would then be \(E A\).
If we want to consider the same problem, but now posed on a general surface, say the planet \(S\), and where \(E\) varies along the surface, then we must consider adding together \(E(\mathbf{x}) \mathrm{d}{S}\). We thus chop our surface into smaller pieces, each with area \(\mathrm{d}S\), multiply each piece with its corresponding energy, and then sum the result.
The result is the surface integral \[ \iint_S E(\mathbf{x}) \, \mathrm{d}S. \] A tutorial for computing surface integrals can be found here.
You will not need to know how to calculate surface integrals in general in the course.