The Divergence of a Vector Field

This section relies heavily on understanding vector fields from . We have separated the details of how the flux density of a region leads to the definition and understanding of the divergence (). The second subsection is optional, but is worthwhile reading for students who are interested in a good conceptual understanding of how divergence is developed.

As we saw in , there are many physical and theoretical representations for vector fields. A natural question is Where exactly is the vector field created? With the vector field in , imagine sketching a curve that follows the direction of the vector field by treating the vectors in the vector field as tangent vectors to your curve. No matter where you start, you should observe that the vector field decreases in strength as you move along the flow. We wish to understand (as a function of position), how much of the vector field is created (or destroyed) at a given location.

We begin this subsection by stating a definition that captures analytically the ideas you reasoned about geometrically in . After the statement of the definition, we discuss what it means.

In other sources you may see the divergence written using a dot product as \(\divg(\vF) = \nabla\cdot \vF\). This notation is very compact and works well with the understanding that the del operator \(\nabla = \langle \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\rangle\) is a function that operates on other functions. However, this notation can also be confusing because of its emphasis on computation rather than conceptual understanding. In this text, we will not generally write the divergence using the del operator.

The divergence of a vector field is a scalar measurement at a point that measures how the strength of the vector field is changing as we look in small neighborhoods around our point. While will show the details for how the divergence is defined, we can make the following qualitative argument: If we are looking at how the strength of the vector field is changing in a small neighborhood of the point \((a,b)\), then we only need to look how fast the horizontal component is changing horizontally and how the vertical component is changing vertically. We make this measurement of changing strength of the vector field by measuring how much of a vector field flows into versus out of a small neighborhood of our point (as was done in ). When looking at a small neighborhood of the point (as shown by the square in ), only the change of the horizontal component of the vector field contributes to flow in or out on the sides and only the change in the vertical component contributes to flow in or out on the top and bottom.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In this subsection, we examine the details of how to measure the density of the creation or destruction of the vector field using a classic calculus approach. Specifically, we will measure how the strength of the vector field changes in a region around a point. Next, using a limit, we examine what happens to our measurement as we shrink the region. Because vector fields change in a continuous fashion, the vector fields don't actually change at a single point. Rather, we will measure the density for the change in strength of the vector field using a limit.

We will develop all of our measurements in a two dimensional setting for now. However, our arguments can be applied to three (or more) dimensions. We start in the same fashion as in . Namely, we will look at how much of the vector field is going into or out of a square centered at a point \((a,b)\). For this development, we will consider a two-dimensional vector field given by \(\vF(x,y)=\langle{F_1(x,y),F_2(x,y)}\rangle\).

We can parametrize the top edge of the box by \(\vr_{\text{top}}(t) = \langle a+t,b+h\rangle\) with \(-h\leq t\leq h\). Similarly, the bottom, right, and left can be parametrized by \[\begin{aligned}\vr_{\text{bottom}}(t) \amp = \langle a+t,b-h\rangle \\ \vr_{\text{right}}(t) \amp = \langle a+h,b+t\rangle \\ \vr_{\text{left}}(t) \amp = \langle a-h,b+t\rangle\end{aligned}\] all of which use parameter values in \(-h\leq t\leq h\).

The amount of the vector field \(\vF\) that is created inside the square around the point \((a,b)\) can be measured by the net amount of the vector field coming into or going out of the square. The amount of vector flow that goes through each of the boundary segments can be measured by looking at just the orthogonal component of the vector field on each particular segment. For instance, on the top segment, the vertical component \(F_2\) determines how much of the vector field goes in or out of the square. Integrating just the vertical component \(F_2\) of the vector field \(\vF\) over the points on the top segment of our square will therefore measure how much of the vector field goes through the top of the square.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.