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Divergence and Curl

Table of Contents:
☞Divergence
☞Curl
☞The Divergence Theorem
☞Stoke's Theorem

Divergence

The divergence of a vector field at a point is defined as the net outward flux of that field per unit volume at that point. If we view the point as a differentially small volume, we can get a feel for what this means.

One way of picturing flux is to treat it like the flow of a liquid. This is not a perfect analogy, but it is useful for visualization. Suppose we have a pool, and in this pool there are two underwater spouts:one supplying water and the other draining the water off. If the volume that we picture does not contain the spout providing water, the net outward flow of fluid from the volume is zero. All of the water that flows in is balanced by water that flows out. However, if our volume contains the spout providing water, the net outward flow of water(the outward flux) will be positive. Similarly, if our volume contains the spout draining off water, the net flow across the surface is negative.

To calculate the divergence of a vector field in Cartesian coordinates, the following method may be used :

    Given a vector field V = (Vx, Vy, Vz)
    div V = del * V = dVx/dx + dVy/dy + dVz/dz

Divergence is useful in electromagnetics because , by utilizing the divergence theorem, we can often express a volume integral (a triple integral) by a surface integral (a double integral) and vice versa, sometimes simplifying our calculations.

In addition, such use of the divergence theorem allows us to express one type of quantity in terms of another. For example, one way to evaluate the free charge in a given volume is to integrate the charge distribution over the entire volume. however, if we are given the D field associated with the charges, the total free charge per unit volume is simply equal to (div D).

Curl

The curl of a vector field is the vector that represents the net circulation of the field. The magnitude of the vector represents the maximum circulation , and the direction(given by the right-hand-rule) is normal to the surface upon which the circulation is greatest.

For example, picture water whirling in a spiral down your sink. Around the center of this small whirlpool there is a net circulation . If the water is spinning counterclockwise, we represent this net circulation as a vector pointing up(from the right-hand-rule). However, if we consider points outside of the center, there is no net circulation around these points. Water simply flows past them, not around them. Thus the curl is zero at points outside of the center.

To calculate the curl of a vectoer field in Cartesian Coordinates, the following method may be used :

    Given a vector field V=(Vx,Vy,Vz)
                               / dVz       dVy      dVx      dVz    dVy    dVx ＼
    curl V = del × V = |----- - ------ ,----- - -----,----- - ----- |
                               ＼ dy        dz         dz       dx       dx      dy   /

A curl-free field is called a conservative field. Such fields have the property that the line integral around any closed loop (often representing the work done in moving a particle) is zero. The same line integral is also independant of path. That is, it depends only upon the endpoints of the path. Thus, conservative fields tend to be very convenient to work with mathematically.

You may not run into the curl of a vector field right away. You will probably start the course studying electrostatic field, which are conservative. Because of this, calculating their curl is not very interesting => it is equal to zero. However, curl will become important as you study magnetic fields. In fact, nearly every equation that contains a dot product in electrostatics has a similar equation using a cross product in magnetostatics.

In the same way that dot products can simplify our calculations, cross products will simplify other calculations. In fact, using Stoke's Theorem, we can often reduce a surface integral (a double integral) to a line integral (a single integral) and vice versa. This may allow us to simplify certain expressions.

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