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In order to establish some properties of the magnetic field, we must review some of the principles of vector calculus. These principles will be our guidance in the next section.
Consider a three dimensional vector field defined by F = (P, Q, R), where P, Q and R are all functions of x, y and z. A typical vector field, for example, would be F = (2x, xy, z2x). The divergence of this vector field is defined as:
diverge
= + + |
Divergence is mathematically significant because it allows us to relate volume integrals and surface integrals, through Gauss' Theorem. Given a closed surface that encompasses a certain volume, this theorem states that:
·da = dv |
The second major concept from vector calculus that applies to magnetic fields is that of the curl of a vector function. Take again our vector field F = (P, Q, R). The curl of this vector field is defined as:
= - , - , - |
Unlike magnetic fields, electric fields never have curls. Recall that the line integral over any closed loop in an electric field is zero, implying that the field cannot "curve" around, as a field with a nonzero curl would.
Just as Gauss' Theorem relates surface integrals and volume integrals using divergence, Stokes' Theorem relates surface integrals and line integrals using curl. Given a closed curve that encompasses a surface,
·ds = ·da |
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