Lecture
All of space is permeated by fields. We live in the gravitational fields of the Earth and the Sun. We are surrounded and permeated by electromagnetic fields. Light is also an electromagnetic field. All around us act the field of atmospheric pressure, the temperature field, etc. So what is a field? A field is any physical quantity that takes different values at different points in space. And what is a physical quantity? It is a quantity that can be measured quantitatively. Mathematically, a field is described by a function or, more generally, by a set of functions of coordinates and time. Thus, a temperature field is a scalar field and is described by a single function of coordinates and time. A gravitational field is a vector field and is accordingly described by three scalar functions of coordinates and time. An electromagnetic field, as we shall see further on, is described by several vector functions of coordinates and time. The structure of fields and the processes occurring in fields are described by partial differential equations, since the independent variables are the spatial coordinates x, y, z and time t. This description of a field is an exact description of it. There is no more precise description of a field than that given by a differential equation. We shall see that the electromagnetic field is described by a system of partial differential equations — Maxwell's equations.
The main characteristic of the scalar field 
is
the vector
where
— the differential operator "nabla," or the Hamilton operator;
grad <p — a vector directed toward the fastest increase
of the function