The paper is based on the mathematical concept of a vector as an element of a linear space whose most common geometric representation is a straight line with specific length and direction. Inherent in this definition is the fact that the concept of a vector is more general than just a straight line with specified length and direction. The term vector is used in this general context in this presentation and goes beyond such particular cases as conventional space vectors ( $$\bar B, \bar D$$ , etc.) and phasors (single frequencyLaplace-transform solutions). Time dependent vectors describing certain integral properties of electromagnetic devices have been used for a long time. Despite their attractivness their usage was rather limited and in a sense isolated. The most widely used field is tensor analysis where the functions relating the vectors have to be, by definition, linear. Some other approaches assume sinusoidally distributed windings, current sheets, radial $$\bar H$$ fields, etc.; all of these drastically idealize the physical picture. No successful attempt has been made in the past to relate directly the terminal properties of devices to the behaviour of the electromagnetic field itself and to relate them toMaxwell's equations without imposing any restriction on the system. The present paper proves that there is a simple mathematical discipline which handles simultaneously and directly the phenomena of the entire electromagnetic field and the terminal performance of electromagnetic devices. This mathematical technique is vectorarithmetic and vector-analysis. The foundation of this treatment uses onlyMaxwell's first two equations and energy relations; thus, unrestricted in any way, it can handle all problems involving the macroscopic electromagnetic field. With this approach some of the old concepts can be revaluated. It turns out, for example that circuits such as equivalent circuits widely used in electric machinery (which describe in the time domain the single-frequency steady state behaviour of one phase) become circuits describing entire machines under far more general operational conditions; as a result parametric amplification techniques, etc., may be directly applied to electric machines and other devices. It is shown in the paper that tensor analysis is more general than usually claimed. The importance of eigenvector transformations is also emphasized. Examples accompanying this introduction to the vector approach are well known problems this is done to demonstrate that most classical problems can be solved directly and with very little effort by using vector functions. Some topics beyond the terms of references set by the title are also discussed and indication is given that network theory, system theory, high frequency devices, etc. all have the same common basis and as such are special cases within this general vector method.