A Brief Review of Vector Calculus

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A Brief Review of Vector Calculus 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 . Divergence of a Vector Field and Gauss' Theorem 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 = (2 x , xy , z 2 x ) . The divergence of this vector field is defined as: diverge = + + Thus the divergence is the sum of the partial differentials of the three functions that constitute the field. The divergence is a function, not a field, and is defined uniquely at each point by a scalar. Speaking physically, the divergence of a vector field at a given point measures whether there is a net flow toward or away the point. It is often useful to make the analogy comparing a vector field to a moving body of water. A nonzero divergence indicates that at some point water is introduced
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Unformatted text preview: or taken away from the system (a spring or a sinkhole). Recall from electric forces and fields that the divergence of an electric field at a given point is nonzero only if there is some charge density at that point. Point charges cause divergence, as they are a "source" of field lines. 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 where the left side is a surface integral over a and the right side is a volume integral. We don't really deal with volume integrals in electricity and magnetism, so some of this theorem is irrelevant. However, when the divergence of a vector field is zero, this equation tells us that the integral through any surface in the field must also be zero....
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