The Divergence in Two Dimensions

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Section 12.2 The Divergence in Two Dimensions

Although the divergence is usually discussed in three dimensions, the construction in Section 12.1 can also be used in two dimensions. The only significant change is that a new type of flux must be defined on curves, rather than surfaces.

If \(C\) is a (simple) closed curve in the plane, it has an inside and an outside. If the outward normal vector is \(\nhat\text{,}\) then the flux of a vector field \(\EE\) out of \(C\) is given by

\begin{equation*} \textrm{flux} = \int_C \EE\cdot\nhat \,ds \end{equation*}

where \(ds=|d\rr|\) denotes infinitesimal arclength.

Applying the argument in Section 12.1 to a small rectangular box yields essentially the same result, namely that the total flux out of the (two-dimensional) box is some sort of derivative, times the area of the box. Thus, in two dimensions we define

\begin{equation*} \grad\cdot\EE = \frac{\textrm{flux}}{\textrm{unit area}} \end{equation*}

and obtain the rectangular coordinate expression

\begin{equation*} \grad\cdot\EE = \Partial{E_x}{x} + \Partial{E_y}{y} . \end{equation*}

When discussing the properties of the divergence, two-dimensional examples are often given, in part because they are easier to interpret (and to draw!). Such examples can always be interpreted using the two-dimensional notions of flux and divergence given in this section, so long as care is taken to note the different units (area vs. volume) in the two- and three-dimensional cases. Alternatively, two-dimensional examples can be interpreted as a horizontal slice of a three-dimensional example, in which the vector field is also horizontal (no \(\zhat\) component), so that there is no flux through the top and bottom faces of the box. In this interpretation, the height of the box does not affect the computation, but nonetheless restores the units of flux per unit volume to the divergence.

The vector fields in Figure 12.2 model this behavior. They appear to be two-dimensional, but are really three-dimensional, as can be seen by clicking on them and rotating the images. (You may wish to zoom out first, using the scroll wheel of your mouse, or by right-clicking and selecting "Zoom to fit".)

Figure 12.2. Interpreting (horizontal) vector fields as two dimensional.