Electromagnetic Fields: Line Integral

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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Finite differences/Lineal Integral

1.3 Finite differences

The differential expressions formulated in this chapter, will also be used for their correspondent interpretation in terms of finite differences

the differential displacements described in expressions such [4] can be approximated with small finite displacements. Further on we will study how small must these steps be in order to obtain an adequate analysis from a discrete point of view.

We will also see how, in general, that the smaller these movements are, more precise will the results be. However, it has a limit related to the format used for the numbers. Diminution larger than certain magnitudes cause, especially in simulations, mistakes due to the truncation of certain quantities. To this we refer in the successiveness as “numeric noise”.

1.4 Line integral.

This definition is very important for the concept of rotational. We are particularly interested in the line integral of a vector function “F” across a path “C”, between points “a” and “b”. This can be expressed the following way:

        (28)

when "a" y "b" are the same point, we talk about a closed integral; in which case we will use the following symbols.

        (29)

Scalar product and defined integral concepts are involved, which we recommended be revised carefully in a mathematical text. For the moment, we remember that being an integral, the limit of an addition, the linear integral “T”, can be approximated as is shown in figure [7].

        (30)

Each element of the addition, in the case of a rectangular system of coordinates will be equal to,

(31)

Of course, the approximation will be better the smaller the longitudes lⁱ are; in other words, the larger the quantity of subdivisions considered between points "a" y "b".

Finally, notice that this operation acts upon a vector function and produces a scalar quantity.


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