Mathematical Physics Vol 1

Chapter 4. Field theory

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denote the function as f ( M ) rather than f ( x 1 ,..., x n ) , as in relation (4.1). When a function is expressed by coordinates, it is said that it is given in analytical form. Examples of scalar fields include the following: temperature, mass, mass density, electric charge, pressure , etc. Since time t can also be one of the variables, we will call a field that does not (explicitly) depend on time a stationary field . Definition The geometric location of the points in which the function f ( M )= f ( x 1 , x 2 , x 3 ) has a constant value C : f ( M )= f ( x 1 , x 2 , x 3 )= C , (4.2) is called an equiscalar surface of a scalar field.

Definition The geometric location of the points in which the function f ( M )= f ( x 1 , x 2 ) has a constant value C : f ( M )= f ( x 1 , x 2 )= C , (4.3) is called an equiscalar line of a scalar field.

4.1.1 Directional derivative. Gradient

The application of mathematical analysis to the study of a scalar field f ( M ) allows for describing its local properties, i.e. changes of f ( M ) when moving from point M to a close point N . Consider a scalar field given by the function f ( x , y , z )= f ( M ) , within a Cartesian coordinate system. It is known that the first partial derivatives of the scalar function f represent the rates of the change of the function in the directions of the coordinate axes. However, it is natural to extend this concept beyond the three directions (in the 3–D space). Extending this idea to include the change of the function in any possible direction brings about the concept of directional derivative of a (scalar) function.

In order to determine this derivative, let us observe a point M in space and a direction through this point, determined by the unit vector e . Let N be a point at this direction, denoted by ℓ , where the distance between M and N is denoted by ∆ s (Fig. 4.1) −→ MN = ∆ s e . (4.4) Let us denote the difference of the function f inpoints M and N by ∆ f = f ( N ) − f ( M ) .

Figure 4.1: Direction of change.