Line of field

In Physical and Mathematical, in order to visualize a vectorial field , one often uses the concept of line of field . It is, at first approximation, the way which one would follow on the basis of a point and following the vectors. More rigorously, a line of field is in any point tangent with the field considered.

From an infinitesimal point of view, the lines of field of a field Φ are the curves directed locally by an element of right-hand side of r' which checks:

$\backslash \; mathrm\; \{det\}\; \backslash \; left\; (\backslash \; mathbf\; \backslash \; Phi,\; \backslash \; mathrm\; D\; \backslash \; mathbf\; R\; \backslash \; right)\; =\; 0$.

The lines of field are orthogonal with the equipotential S.

A certain number of quantities, like the Rotational or the Divergence in a point, can thus be “observed”. So applications of the lines of field, like those of the potential of Douady-Hubbart for the whole of Mandelbrot, remain purely theoretical; the lines of field can present an interesting physical interpretation, in particular in plasma physics.

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