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Section 14.5 Exploring the Divergence in Polar Coordinates

Figure 14.5.1 below shows the relationship between flux and divergence using polar coordinates and basis vectors. You can choose the vector field \(\boldsymbol{\vec{v}}\) by entering its components \(v_r\) and \(v_{\phi}\text{,}\) move the box by dragging its center, and change the size \(s\) of the box by moving the slider.

Figure 14.5.1. The relationship between flux and divergence.

Activity 14.5.1. Exploring Divergence.

Enter the (two-dimensional vector field of your choice into the applet in Figure 14.5.1 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 \(r\text{,}\) then try more complicated functions.