Exploring the Curl in Polar Coordinates

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Section 12.12 Exploring the Curl in Polar Coordinates

Figure 12.13 below shows the relationship between circulation and curl using polar coordinates and basis vectors. 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 12.13. The relationship between circulation and curl.

Activity 12.5. Exploring Curl.

Enter the (two-dimensional vector field of your choice into the applet in Figure 12.13 by entering its components. Determine the circulation 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 $\frac{\textrm{circulation}}{\textrm{area}}$) with the computed value of the curl at the center of the box (shown as $\grad\times\vv\big|_P$).

What do you notice?

Hint.

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