Activity 13.6.2. Graphical Representations of Divergence.
(a)
(i)
Draw a circle in the first quadrant of the vector field \(\vF\) depicted in Figure 13.6.12. Based on the flow of the vector field into or out of the circle, do you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in the first quadrant?
(ii)
As you move along the flow of the vector field in the first quadrant of Figure 13.6.12, does your vector field increase in magnitude, decrease in magnitude, or have constant magnitude?
(iii)
Draw a circle in each of quadrants II, III, and IV. Based on the flow of the vector field into or out of your circles, do you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in quadrants III, II, and IV?
(iv)
As you move along the flow of the vector field in the third quadrant of Figure 13.6.12, does your vector field increase in magnitude, decrease in magnitude, or have constant magnitude.
(v)
Based on your arguments above, describe why the divergence of \(\vF\) is negative for all points in the \(xy\)-plane.
(b)
Look at the plot of the vector field \(\vG\) in Figure 13.6.13 and state whether you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in each of the four quadrants. You can make your argument in terms of the change in magnitude along the flow of the vector field or in terms of the net flow into or out of a small region on the plane. You may need to make separate arguments for each of the four quadrants.
A vector field in which vector magnitudes increase as distance from the origin increases. Vectors are oriented as if they follow hyperbolas with asymptotes \(y=x\) and \(y=-x\text{.}\) Vectors above both asymptotes or below both asymptotes result in counterclockwise rotation. The other vectors result in clockwise rotation.
(c)
Look at the plot of the vector field \(\vH\) in Figure 13.6.14 below and state whether you think the vector field is increasing in strength, decreasing in strength, or not changing in overall strength in each of the four quadrants. You can make your argument in terms of the change in magnitude along the flow of the vector field or in terms of the net flow into or out of a small region on the plane. You may need to make separate arguments for each of the four quadrants.
A vector field having longer vectors where \(x \lt 0\text{.}\) For \(x>0\text{,}\) vectors appear to get longer as distance from the \(x\)-axis increases.
