Exploring the Divergence

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Section 12.3 Exploring the Divergence

Figure 12.3 below shows the relationship between flux and divergence in two dimensions. You can choose the vector field $\boldsymbol{\vec{v}}$ by entering its components $v_x$ and $v_y\text{,}$ move the box by dragging its center, and change the size $s$ of the box by moving the slider.

Figure 12.3. The relationship between flux and divergence.

Activity 12.1. Exploring Divergence.

Enter the (two-dimensional vector field of your choice into the applet in Figure 12.3 by entering its components. Determine the flux per unit area at several locations by moving the box and adjusting the slider. In each case, compare your result (shown in the applet as $\frac{\textrm{flux}}{\textrm{area}}$) with the computed value of the divergence at the center of the box (shown as $\grad\cdot\vv\big|_P$).

What do you notice?

Hint.

Start with vector fields whose components are linear functions of $x$ and $y\text{,}$ then try more complicated functions.