Exploring the Curl

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Section 14.10 Exploring the Curl

Figure 14.10.1 below shows the relationship between circulation and curl 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 14.10.1. The relationship between circulation and curl.

Activity 14.10.1. Exploring Curl.

Enter the (two-dimensional vector field of your choice into the applet in Figure 14.10.1 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 $x$ and $y\text{,}$ then try more complicated functions.