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The book develops the fundamental ideas of the famous Kac-Rice formula forvectorvalued random fields. This formula allows to compute the expectation andmoments of the measure, and integrals with respect to this measure, of the setsof levels of such fields. After a presentation of the historical context of theKac-Rice formula, we give an elementary demonstration of the co-area formula.This formula replaces the change of variable formula in multiple integrals anda direct application of this formula gives the Kac-Rice formula for almost alllevels. We emphasize the necessity of having the formula for all levels,because for some applications one needs, for example, the formula for levelzero.

Kac-Rice formula: A contemporary overview of the main results and applications