Coils are one of the basic elements employed in devices. They are versatile, in terms of both design and manufacturing, according to the desired inductive specifications. An important characteristic of coils is their bidirectional action; they can both produce and sense magnetic fields. Referring to sensing, coils have the unique property to inductively translate the temporal variation of magnetic flux into an AC voltage signal. Due to this property, they are massively used in many areas of science and engineering; among other disciplines, coils are employed in physics/materials science, geophysics, industry, aerospace and healthcare. Here, we present detailed and exact mathematical modeling of the sensing ability of the three most basic scalar assemblies of coaxial pick-up coils (PUCs): in the so-called zero derivative configuration (ZDC), having a single PUC; the first derivative configuration (FDC), having two PUCs; and second derivative configuration (SDC), having four PUCs. These three basic assemblies are mathematically modeled for a reference case of physics; we tackle the AC voltage signal, V (t), induced at the output of the PUCs by the temporal variation of the magnetic flux, Φ(t), originating from the time-varying moment, (t), of an ideal magnetic dipole. Detailed and exact mathematical modeling, with only minor assumptions/approximations, enabled us to obtain the so-called sensing function, F, for all three cases: ZDC, FDC and SDC. By definition, the sensing function, FSF, quantifies the ability of an assembly of PUCs to translate the time-varying moment, (t), into an AC signal, V (t). Importantly, the FSF is obtained in a closed-form expression for all three cases, ZDC, FDC and SDC, that depends on the realistic, macroscopic characteristics of each PUC (i.e., number of turns, length, inner and outer radius) and of the entire assembly in general (i.e., relative position of PUCs). The mathematical methodology presented here is complete and flexible so that it can be easily utilized in many disciplines of science and engineering.