Particle accelerators are devices for increasing the velocity of particles. Their core functioning lies in electromagnetic interactions, and the phenomena observed in and because of them have diverse applications in biology, chemistry, industry, and particle physics. This study dives into the historical improvements in particle accelerators, going through what has made it possible to go from the basic functioning of the Cockcroft-Walton and Van de Graff accelerators to synchrotrons and storage rings like the Large Hadron Collider (LHC), the most powerful particle accelerator ever built. Furthermore, it explores the limitations of each stage of these machines. To facilitate further research on this current topic, it is also necessary to examine the mathematical tools used in the study of the electromagnetic, relativistic, and quantum phenomena observed in particle accelerators. The essential tools described in this first study are related to multivariable calculus, starting with functions and vectors, followed by their applications in multivariable calculus with multivariable functions and partial derivatives, and concluding with vector fields, gradient vector fields, vector field’s potential, line integrals, vector field divergence, and vector field curl. A complete understanding of these mathematical concepts will allow for further research to understand Maxwell’s Equations and Lorentz’s force law, which require the operation of the studied concepts applied to open and closed surfaces.