Divergence

Let us define a vector function of two variables. This function might represent any number of things, e.g. velocity and direction of particles in a fluid, the magnitude and direction of the electric field in a planar region of space, or the flow of heat in a sheet of metal.

In[214]:=

  vF1[x_,y_] := {Sin[x],Cos[y]}

We can examine this function in various ways. We can draw vectors representing the field. See the vector fields section.

In[215]:=

  PlotVectorField[vF1[x,y],{x,-5,5},{y,-5,5},Axes->True];

Notice that there are points in this field where the vectors seem to be converging (such as the point x ť 3.1, y ť 1.5) and other points where the field appears to be diverging (such as the point x ť 0, y ť 1.5). If this field represents fluid flow, then the areas where vectors are converging can be considered sinks and the areas where the lines are diverging can be thought of as sources. In other words, particles flow from the sources to the sinks. But how do we measure the strength of a source or a sink? That's where the divergence is useful.

The divergence of a vector function f(x,y) is written
śfx / śx + śfy / śy.

We can calculate the divergence of vF1 symbolically.

In[216]:=

  divF1 = div2[vF1][x,y]

Out[216]=

  Cos[x] - Sin[y]

Note that this is a scalar function of x and y. Now let's plot the divergence over the same region we examined above.

In[217]:=

  ContourPlot[divF1,{x,-5,5},{y,-5,5}];

Note: Lighter regions are higher.

In[218]:=

  Plot3D[divF1,{x,-5,5},{y,-5,5},AxesLabel->{"x","y","z"}];

The more positive the divergence, the greater the source strength. The more negative, the stronger the sink. When the divergence is zero, the point is neither a source nor a sink. We can say that there is no change in vector magnitude about a point with zero divergence.

Another example

More Examples

Up to Divergence and Curl