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====== Differences ====== This shows you the differences between two versions of the page.

--- gibson:teaching:spring-2015:math445:lab11 [2015/04/13 13:07]
gibson
+++ gibson:teaching:spring-2015:math445:lab11 [2015/04/15 19:36] (current)
gibson
@@ Line 20: / Line 20: @@
-**Problem 3.** Superimpose on your previous plot the path of a raindrop that falls at $(x,y) = (0.9,0.9)$.
+**Problem 3.** Superimpose on your previous plot the path of a raindrop that falls at
+$\vec{x} = (x,y) = (0.9,0.9)$. We're assuming the velocity of the raindrop as it drips down is proportional to the negative of the gradient of the height. That is, $\vec{v} = -c \vec{\nabla} h $,
+where $\vec{v} = d\vec{x}/dt$ is the $x,y$ velocity of the raindrop. For convenience set $c=1$ (this won't change the path). Then compute the path of the raindrop numerically using //forward Euler time-stepping//,
+\begin{eqnarray*}
+\vec{x}(t+\Delta t) = \vec{x}(t) + \Delta t \, \vec{v}(\vec{x})
+\end{eqnarray*}
 {{:gibson:teaching:spring-2015:math445:montesol_path.png?direct&400|}}
-**Problem 4.** Then show the path of the raindrop in 3d, superimposed on the surface plot.
+**Problem 4.** Then show the path of the raindrop in 3d, superimposed on the surface plot. You will need to use Matlab's ''plot3'' function.
 {{:gibson:teaching:spring-2015:math445:montesol_path3d.png?direct&400|}}
+**Problem 5.** Recompute the plot of problem 3 using Matlab's ''ode45'' numerical integration function instead of forward Euler.
+**Problem 6.** Plot the paths of 100 raindrops that fall randomly on the mountains. Show the hundred raindrop paths on both the contour plot and the 3d surface plot.

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